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)?