I have now been staring at this particular sentence for way too long, and thus will start using any and all communication lines I can find to get assistance. Either I'm being way too stupid, or the author neglects to mention some salient detail.

Setup: [tex]\phi\colon G^\prime\to G[/tex] is a group homomorphism, [tex]A\in kG-\operator{mod}[/tex], [tex]A^\prime\in kG^\prime-\operator{mod}[/tex]. [tex]A[/tex] can be given the structure of a [tex]kG^\prime[/tex]-module by pulling back through [tex]\phi[/tex], i.e. we define [tex]g^\prime a:=\phi(g^\prime)a[/tex] for [tex]g^\prime\in G^\prime[/tex] and [tex]a\in A[/tex].

So far it's all crystal clear for me. However, it then turns out that we're highly interested in using a morphism [tex]f\in\operator{Hom}_G(A,A^\prime)[/tex] and I cannot for the life of me find out how such beasts are guaranteed to exist. If it where [tex]f\in\operator{Hom}_{G^\prime}(A,A^\prime)[/tex], I wouldn't have any problems with it; but then the stuff I need/want to do with it don't work out.