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I have a trouble in Fragment life cycle and need a resolution to come out of it

I have done ample research on this, and there is not one clear solution on the problem.
In the life-cycle, particularly in the Fragment life-cycle, following Exception comes any moment after onPause().
java.lang.IllegalStateException: Can not perform this action after onSaveInstanceState
My logic says, that to continue with the current fragment, after it reaches this state, I have to restart the activity and again point back to the intended fragment using Intent.
I want to be clear on what is happening and what should be real solution to deal with it.
I need to know the pros and cons of this mechanism; its importance in Fragment or Activity life-cycle.
Also, if I am changing the Windows Feature in onCreate to not to go to sleep, unless if the user has manually pressed the home button, will still the activity will go to this state?

This exception happens when you're trying to add/remove/replace/interact in any other way with a Fragment inside the Activity when it's paused.
Which means Activity will not be able to restore it's state (restore the state of a Fragment which has been changed) if it will be destroyed right away.
Best solution here, is to check that Activity is NOT paused during the interaction with a Fragment.
Another option is to use commitAllowingStateLoss() to interact with Fragment transaction, with a risk of losing it's state.
See:
https://developer.android.com/reference/android/app/FragmentTransaction.html#commitAllowingStateLoss()
In a perfect world you should analyze each crash carefully and add checks to verify that you interact with fragments only when Activity is up and running.

A better explanation is presented in a new Android developer reference and guide documents for using JetPack Life Cycle Listener.
https://developer.android.com/topic/libraries/architecture/lifecycle#kotlin
The library makes the components Activity Life Cycle aware. That means you do not require an abstract baseActivity class which overrides every life cycle callback, and record that state in a boolean variable. LifeCycle listener will do it for you.
All you have to do is stop introducing a new fragment or stop any Loader that updates the UI when its response returns. The right time to do this is before onStop or onSavedInstance state is called, and your components will be made aware of it.
It clearly states that after the onSavedInstancState or onStop is called the UI becomes immutable till the onStart of the Activity is called again. Sometimes you have to call restart the same activity using NEW TASK and CLEAR TASK flags using intent, when this state occurs and there is no chance that otherwise onStart is going to be called.
Happy Coding :-)

what happens to an activity when in pause state?

When an Activity goes in pause state, an instance of it remains in the Activity stack that is managed by operating system. And I know after a while, it destroys the instance. Now my question is whether the onDestroy() method is called when operating system destroys the Activity instance after a long while?
Like if I put a Toast inside onDestroy() method, that will be shown when the Activity instance gets destroyed by OS? (I know it will be shown by pressing back button).
The nature of this question makes it hard to test because sometimes it takes a day or more for OS in order to destroy an Activity instance in stack.

There is no guarantee that it will be called. You can see Activity#onDestroy for that.
Note: do not count on this method being called as a place for saving data! For example, if an activity is editing data in a content provider, those edits should be committed in either onPause() or onSaveInstanceState(Bundle), not here. This method is usually implemented to free resources like threads that are associated with an activity, so that a destroyed activity does not leave such things around while the rest of its application is still running. There are situations where the system will simply kill the activity's hosting process without calling this method (or any others) in it, so it should not be used to do things that are intended to remain around after the process goes away.
By simply killing it could be e.g. a System.exit call or something similar where it skips the lifecycle hooks.

Activity's onDestroy() method not called after removing activity from background by swiping

I have implemented ActivityLifecycleCallbacks in a private class inside my Application class.
If Home button or Back button is used, lifecycle's onStop() and onPause() methods are getting called. If I kill the activity by swiping out from background, onDestroy() method is not getting called. It gets called when I start my activity next time. I am using moveTaskToBack(true) in case of Back press.
What must be the issue and which lifecycle method should get called when we swipe out from background?

From the documentation of onDestroy():
Note: do not count on this method being called as a place for saving
data! For example, if an activity is editing data in a content
provider, those edits should be committed in either onPause() or
onSaveInstanceState(Bundle), not here. This method is usually
implemented to free resources like threads that are associated with an
activity, so that a destroyed activity does not leave such things
around while the rest of its application is still running. There are
situations where the system will simply kill the activity's hosting
process without calling this method (or any others) in it, so it
should not be used to do things that are intended to remain around
after the process goes away.
Such a situation is swiping the app out of the recent tasks list.
Check out this answer on how to get notified when the app is getting swiped out of Recents.

I remember reading a similar question on StackOverflow. Here was the top answer:
Handle exit application from Task Manager
Handle exit application from Task Manager
Unfortunately there isn't a good answer for handling this situation. If the app is force-killed, onDestroy method isn't necessarily called. According to the documentation.
Note: do not count on this method being called as a place for saving data! For example, if an activity is editing data in a content provider, those edits should be committed in either onPause() or onSaveInstanceState(Bundle), not here.
If you can, clean up in the onPause() method. In order for the user to get to that screen to kill the app, it has to have been backgrounded and thus onPause() would be called. (see documentation)

swiping app from back ground i.e. clearing from recents causes the whole process and task to be killed and that time onDestroy() may not be called , when user presses back from an activity then only that activity is killed not the process.

Pausing/Resuming Activity outside of an activity

I have 2 activities. Main Activity A & Activity B
I do not want Activity A to destroy. I am starting Activity B in a new task.
public static void startActivity(Class<?> startClass) {
Intent intent = new Intent(Constants.getActivity(), startClass);
intent.addFlags(Intent.FLAG_ACTIVITY_NEW_TASK);
Constants.getActivity().startActivity(intent);
}`
Constants.getActivity() returns the Context on current activity
startClass is the either activity "A" or activity "B"
Thing is they create/destroy the activities and they leak. Am I doing it wrong? How can I start activity "B" from activity "A" and vice versa keep them both in background when I dont need them.

First of all, what are you trying to do? You should always separate things you want to do in the background from your UI. Think of your Activities are simply a container to show the UI, everything else can be stored and restored from persistent storage or savedinstance bundles.
It is very crucial that you understand the difference between Activity lifecycle and Service lifecycle here.
I'm going to refer to my answer from another question here:
Your app will be in the background, onResume and onPause will both be called, provided that the OS have enough memory to keep the new app and all the old apps.
If you have a long running process that you need while the user not looking at it, use a service.
If you need the user to return the app in the same state, you need to do the work in onResume and onPause to save the state information and initialize your UI again when the user comes back. If you are really worried that it can get killed (well, it shouldn't lose the bundle I think), you can store them in SharePreferences for your app.
If you want to know when the app returns from that specific share intent, use startActivityForResult

You cannot keep an activity "alive" as you said. When an activity is paused, Android system is always able to claim its memory and unload it, at any time.
But if you want to switch from A to B, then maybe A and B can be different views inside a single activity. Maybe you'll want to have a look at http://developer.android.com/reference/android/widget/ViewFlipper.html

When you use tasks, cleanup is very important. You need to cleanup all tasks in the activity. readthis
If the activity does not have a lot of crazy initialization, just use finish and onCreates. Else be aware that onResume will be called often as you switch between activity's. Cleanup will be crucial here. If you dont cleanup, its possible one of your activities (with dead object reference from the other cleaned up activity) may come back up from the activity stack and throw exceptions. Its very difficult to debug this kinda exception.

Want to start a new activity but let the original activity finish first

I have an Android activity we'll call A which has a button and another activity B. When the user clicks the button in Activity A, I'd like to finish A (let both onStop and onDestroy finish running) and then start up the instance of B. When I put a finish() and startActivity() call in the button click listener, the instance of B starts up before the old instance of A finishes. Can someone help me figure out a way to do what I'm looking for?

What you are looking for is not possible and actually is against Android's activity lifecycle implementation.
Correction
It is possible with android:noHistory="true" tag in your manifest, but for what you are trying to do it seems wrong (read the EDIT)... Messing with the activity stack makes a non intuitive application!
Android OS doesn't let you control when activities will be removed from memory (or killed), and therefore all these fancy "Task killers" are so popular (DONT use them, they only make things worse).
When your activity's onStop() is being called, the activity stops completely, and it just hangs in your memory, but that's fine...
If you want to reset the state of activity A, or close the app when exiting activity B, just create a set of rules in both onResume() and onStop(), you can do everything you wish by creating a set of rules in those functions.
for example: have a boolean in activity A that turns true just before calling activity B,call finish() on your activity A's if this boolean is true
I suggest that you take a look at Android's Activity lifecycle diagram, and make sure that everything you do follows the best practice.
EDIT
I saw your comment, it seems like you are trying to create things that are already in your memory, don't recreate them, it's a waste of CPU time, memory, and battery.
Instead, create a static class with a singleton that will hold all your shared data !

I believe you're looking for
onPause()
which is what gets called when the activity is sent to the background. You can do whatever cleanup you want in there. onStop should only be called when a user is exiting out of your program (or launching another one)

onPause is a better place to do this cleanup. See the Saving Persistent State section of the Activity doc.
When an activity's onPause() method is called, it should commit to the backing content provider or file any changes the user has made. This ensures that those changes will be seen by any other activity that is about to run. You will probably want to commit your data even more aggressively at key times during your activity's lifecycle: for example before starting a new activity, before finishing your own activity, when the user switches between input fields, etc.
While I'm not definite that your cleanup is for user changes, the bold sentence above implies that onPause will complete before the next Activity is created. Of course that probably implies that you'll have to move some setup to onResume...
Alternatively, you could move all your cleanup code to a method, let's just call it cleanup and then just call it before you start activity B. You'll have to put in appropriate guards for your onDestroy cleanup too of course.

override finish() method.
implement cleanUp() method.
create boolean isClean=false in the activity
in cleanUp() write your clean up code.
call cleanUp() in your finish()
check for isCleaned in finish() or in cleanUp() if its true then ignore the clean
now before you start B , call cleanUp() and set isCleand=true
after you call B , call finish()

Start activity A
from inside A startService(c) and finsh A
from inside the service , start Activity B