Retain Fragment state between Activities

Its possible to retain a Fragment between Activities?

Lets say I have Activity A with Fragment F_Left placed at the left and Fragment F_Right placed at the right. If I want to launch a new Activity and keep Fragment F_Left... how can I do it?

Can I retain Fragment F_Left state between activities?

Note that I want to launch a new Activity because Fragment F_Left is my app menu and Fragment F_Right changes completely the context of the user operations... and my app have many of operations, so it makes sense to have an Activity per operation.

I know its possible to retain Fragment within an Activity, but as Fragment life cycle is closely tied to the container Activity I don't know if this is possible keep Fragment state between Activities.