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Move from one Fragment inside Activity to another Fragment inside Activity

So I have Activity A with Fragment A.1, and I also have Activity B with Fragment B.1.
What I want to ask is, how do I move directly from Fragment A.1 to Fragment B.1?
I know to move from Fragment A.1 to Activity B, is by:
Intent i = new Intent (getActivity (), MainActivity.class);
startActivity (i);
getActivity ().finish();
But how to move straight to Fragment B.1?
Each Activity A and Activity B has a different <FrameLayout> for Fragment replacement
UPDATE 1.0
I've tried my own way and also the way #cewaphi answered with code like this,
In Activity A:
Intent i = new Intent(TransactionDone.this, MainActivity.class);
i.putExtra("immediatelyTransactionToFragment", true);
startActivity(i);
finishAffinity();
In Activity B:
boolean shouldTransitionToFragment = getIntent().getBooleanExtra("immediatelyTransationToFragment", true);
if (shouldTransitionToFragment) {
Fragment fragment = new Wallet();
FragmentTransaction transaction = getSupportFragmentManager().beginTransaction();
transaction.replace(R.id.mainFrameLayout, fragment);
transaction.commit();
Log.d("DEBUGGING REDIRECT", "Go to Fragment B.1");
}
The log "Go to Fragment B.1" was created but the transaction doesn't work

When using a single activity and e.g. using the navigation component is not an option.
Consider the following:
In your fragment A.1 when starting the activity store a Boolean
i.putExtra("immediatelyTransationToFragment", true);
In activity B
shouldTransitionToFragment = getIntent().getBooleanExtra("immediatelyTransationToFragment");
// after activity was created
if (shouldTransitionToFragment) {
// Execute the transition action as you would when pressing the button
}
Update 2020/09/07
You are trying to transition to a Fragment from your Activity like this:
FragmentTransaction transaction = getSupportFragmentManager().beginTransaction();
transaction.replace(R.id.mainFrameLayout, fragment);
transaction.commit();
The documentation states that you should first add the fragment to the activity:
ExampleFragment fragment = new ExampleFragment();
fragmentTransaction.add(R.id.fragment_container, fragment);
fragmentTransaction.commit();
Apparently your transition works when you click your button. Are you doing it the same?
But I assume at that time the activity has already been created.
Try once to add your fragment instead of replace. I don't know how your container is initialised but adding might be the operation you want, I refer to this good answer for clarification.
Also consider to perform to call this transaction after your activity was created.

How to intent from activity to specific fragment?

I want to send an intent from one activity to a specific fragment like this picture, and fragment is in the fragmentactivity (fragmentactivity includes five fragments). I don't have any idea to implement it.

I don't exactly know what do you mean by "to intent from activity to specific fragment". So I just assume you want to show/navigate to your fragment in your activity. Here's the simple code:
//show your fragment inside your activity
YourFragment fragment = new YourFragment();
getIntent().putExtra("KEY", "Value"); //pass data to your fragment
transaction = this.getSupportFragmentManager().beginTransaction();
transaction.replace(R.id.IDofYourLayoutToBeReplaced, fragment);
transaction.commit();
//in your fragment, retrieve the data
String yourVariable = getActivity().getIntent().getExtras().getString("KEY");
or to better understand, you can check this link

Android Fragments sharing info ways

i have a internal discussion about what way is better to share info between fragments contents inside a controller activity. In a first classical way, you can set arguments when you are going to replace fragments as follows:
//Just now i'm inside Fragment 1 and i'll navigate to Fragment 2
Fragment newFragment = getFragmentManager().findFragmentByTag(Fragment2.TAG);
Bundle b = new Bundle();
b.putBoolean("test1", true);
// Create new fragment and transaction
if(newFragment==null)
newFragment = Fragment2.newInstance(b);
FragmentTransaction transaction = getFragmentManager().beginTransaction();
transaction.setTransition(FragmentTransaction.TRANSIT_FRAGMENT_OPEN)//.setCustomAnimations(R.anim.enter_anim, R.anim.exit_anim)
.replace(R.id.fragment_place, newFragment, Fragment2.class.getName())
.addToBackStack(newFragment.getClass().getName())
.commit();
The newInstace method does as i meant above, so, with setArguments:
public static Fragment2 newInstance(Bundle arguments){
Fragment2 f = new Fragment2();
if(arguments != null){
f.setArguments(arguments);
}
return f;
}
But Fragment1 and Fragment2 they are both inside a ControllerActivity, so i can also think about a second way to share information obtained in Fragment1 towards Fragment2, through declaring attributes in the ControllerActivity, so i could do (declaring previously an object in the activity) as follows inside any fragment:
EDIT
public class ControllerActivity extends FragmentActivity{
int value = 5;
...
And then, inside my fragment:
((SplashActivity)getActivity()).value = 10; //i can assign or recover value when i desire
My question is what inconveniences would have doing as the second way.

Writing code using 2nd way is fast. But the problem is you have to cast the general Activity to the more specific SplashActivity in which the value variable exists. If you want to use the Fragment with another Activity, or you want a Fragment to be a general purpose UI component you have to use interface for passing the data.
As mentioned in comments, bellow links provide more details about interface/callback method:
android docs
video from slidenerd
Hope this answers your question.

Android navigating to parent Fragment Activity from within Fragments

The main fragment activity in my application has the following function
private final void insertFragmentIntoView(final SherlockFragment fragment,
String tag) {
FragmentManager fm = getSupportFragmentManager();
FragmentTransaction ft = fm.beginTransaction();
ft.add(R.id.fragment_content, fragment, tag);
ft.commit();
}
The R.id.fragment_content is a frame layout and i basically insert a new fragment into this frame. Now the frame that i insert has a button that should take me onto a new screen. What i want is that all navaigation in my application should take place in my FragmentActivity. How can i call my FragmentActivity from with a child fragment ?
Kind Regards

Make a listener class in your fragment and your parent activity will implement that listener .
Now register listener in your fragment and call method in which you want to perform some action.
More you can see this link http://developer.android.com/training/basics/fragments/communicating.html
Example how fragment and activity communicate.
class MyFragment extends Fragment{
class interface MyFragmentListener {
doSomeAction();
}
MyFragmentListener myListener;
onAttach(){
myListener=(MainActivity )getActivity();
}
onButtonClick(){
myListener.doSomeAction();
}
}
class MainActivity extends FragmentActivity implements MyFragment.MyFragmentListener
{
doSomeAction(){
//TODO perform some action from your fragment to activity
}
}

Using listeners is the recommended way of communicating between Fragment and your activity.
See this Android documentation section for infromation. Long story short they just implement a listener interface by the Activity class and cast getActivity() result in a fragment to a listener.
From my personal experience this is very convenient because lets you to:
Easilly switch underlying activity (e.g. you host entire fragment in a wrapper activity for compatibility in pre-3.0 and host this fragment along with others in 11+)
Easilly control if the wrapper activity supports callbacks or not. Just check is it does implement the listener and do your app specific actions if it doesn't.

Transition to FragmentActivity

I can transition to a Fragment class using the code below, but how do i transition to a FragmentActivity, using the same code.
FragmentActivity activity = new ABC(); // ABC is a FragmentActivity
FragmentTransaction fragmentTransaction = getFragmentManager().beginTransaction();
fragmentTransaction.replace(R.id.details, activity); // replace errors out
fragmentTransaction.commit();

You can't replace a view or a fragment with a FragmentActivity! A FragmentActivity is just an Activity and has been created for Android Compatibility support.
To summarize, an Activity can contains Fragments which can be dynamically replaced. The only way to navigate between Activities is by intent.