Given a group G and a normal subgroup N of G, is there an action of G on N such that, whenever g,h are distinct members of the same N-coset, we have g•n≠h•n? If not, then can this be done in the case G is abelian?

Take for granted that we may select a set of coset representatives for the N-cosets to use as "origins" for each N-coset. I'm working on some abstract analysis/descriptive set theory ideas, and got stuck on this thought because the algebra got a little too far out of my element. If it can't be done, a counterexample would be GREATLY appreciated.

Given a group G and a normal subgroup N of G, is there an action of G on N such that, whenever g,h are distinct members of the same N-coset, we have g•n≠h•n? If not, then can this be done in the case G is abelian?

Take for granted that we may select a set of coset representatives for the N-cosets to use as "origins" for each N-coset. I'm working on some abstract analysis/descriptive set theory ideas, and got stuck on this thought because the algebra got a little too far out of my element. If it can't be done, a counterexample would be GREATLY appreciated.

Okay. Gotta update this question: In this setting, the groups are all either countably infinite or of size continuum.

Also it's important to actually be able to describe the action.