I have written a module and in order to run the main code where the module is imported needs to have the language extension {-# LANGUAGE OverloadedStrings #-} or the option needs to be given when things are compiled of course.
If I include the {-# LANGUAGE OverloadedStrings #-} in the module this is no cure and the user will still need to do the same in her code or when compiling.
Is there any way that I can do “something” in the actual module in order to lift this problem from the end user and so to speak “hide” the required language extension?
No, your library exposes features that require language extensions to be used. This is something you should state in your documentation and examples. There’s no way to turn it on on user code.