The $n$th group cohomology is the $n$th right derived functor of the left exact $M \mapsto M^G$ functor. Using the equivalence $\mathbb Z[G]$-$\mathrm{Mod} \cong G$-$\mathrm{Mod}$ we can show that $\mathrm{Hom}_{\mathbb Z[G]}(\mathbb Z, -)$ is the "same" functor as $M \mapsto M^G$. In particular, $H^i(M; G) \cong \mathrm{Ext}^i_{\mathbb Z[G]}(\mathbb Z, M)$.
Now we know that $\mathrm{Ext}^i_{\mathbb Z[G]}(\mathbb Z, M)$ can be obtained as a left derived functor of $\mathrm{Hom}_{\mathbb Z[G]}(-, M)$ evaluated at $\mathbb Z$.
My question is: is there some nice description of the functor $\mathrm{Hom}_{\mathbb Z[G]}(-, M)$ with group actions? Is it related to the "module of coinvariants" functor?