The fact that you instantiate the component in the base class doesn’t mean that you inject all the dependencies into the base class.
In fact, it is not just bad decision, but such a design decision would result in a maintenance hell. I would never propose anything of this kind.
Of course you have a separate “inject” method for each specific Fragment.
I open sourced my own application and you can see the approach I use: https://github.com/techyourchance/idocare-android