Let $\Gamma$ be a group acting freely and cocompactly on an acyclic space $X$. I know that there is a isomorphism $H^*(\Gamma, \mathbb{Z\Gamma}) \cong H^*_c(X; \mathbb{Z})$ of $\Gamma$-modules. The proof I've seen in Brown's book uses the structure of $\mathbb{Z}\Gamma$ heavily, to give a natural isomorphism at the cochain level. I have two questions.
Suppose $M$ is a $\Gamma$-module. Is there any such (if not so strong) relation between $H^*(\Gamma,M)$ and the (compactly supported) cohomology of $X$? I would guess it would be something relating $H^*(\Gamma,M)$ and $H^*_c(X, M_\Gamma)$. Here $M_\Gamma$ is the group of coinvariants of $M$ under the $\Gamma$-action.
Is there any homological counterpart to this? Specifically can we say something about $H^{lf}_*(X)$ in terms of $H_*(\Gamma)$ with some coefficients?
Maybe a simpler case for 1 would be if we have a group homomorphism $\varphi: \Gamma \to \Lambda$, and consider $M= \mathbb{Z}\Lambda$, with the $\Gamma$-action via $\varphi$.