I'm learning some group cohomology from the third section in Serre's Local Fields, and I'm up to the section on change of group. If f:G'\rightarrow G is a homomorphism of groups and $A$ is a $G$-module, there is an induced G'-module structure given by s'\cdot a=f(s')\cdot a, for s'\in G' and $a\in A$ (this is Serre's notation, I would reserve G' for the commutator subgroup, but oh well). This induces both maps on cohomology H^q(G,A)\rightarrow H^q(G',A) and homology H_q(G',A)\rightarrow H_q(G,A).
My question is: If $A$ is a projective/injective/relatively projective/relatively injective $G$-module, must $A$-as-a-$G'$-module be such a G'-module? If not, are there any assumptions we could make about f:G'\rightarrow G so that this would be true (e.g., $f$ surjective)?