Let $\alpha : G\rightarrow G'$ be a group homomorphism.
There is a natural functor $\alpha^\# : Mod(G')\rightarrow Mod(G)$ sending a $G'$-module $M$ to the $G$-module given by the same underlying abelian group as $M$, with $G$-action defined via $\alpha$.
This functor is exact, but it doesn't obviously preserve projectives or injectives (at least the hypotheses of the adjoint functor criterion does not apply here).
However, it is stated in Ken Brown's book "Cohomology of Groups" (II.6), that $\alpha^\#$ sends $G'$-projectives to $G$-modules which are acyclic for group homology.
Is this clear? I don't see why.
As requested, here is the full statement:
"Given a homomorphism $\alpha : G\rightarrow G'$ and projective resolutions $F$ and $F'$ of $\mathbb{Z}$ over $\mathbb{Z}G$ and $\mathbb{Z}G'$ respectively, we can regard $F'$ as a complex of $G$-modules via $\alpha$. Then, $F'$ is acyclic (although not projective, in general, over $\mathbb{Z}G$), so the fundamental lemma I.7.4 gives us an augmentation-preserving $G$-chain map $\tau : F\rightarrow F'$, well defined up to homotopy. The condition that $\tau$ be a $G$-map is expressed by the formula $\tau(gx) = \alpha(g)\tau(x)$ for $g\in G,x\in F$. Clearly $\tau$ induces a map $F_G\rightarrow F'_{G'}$, well-defined up to homotopy, hence we obtain a well-defined map $\alpha_* : H_*(G)\rightarrow H_*(G')$"