I understand that as abelian groups $C_i(X,M)$ are same as $C_i(X,\mathbb{Z})\otimes_{\mathbb{Z}}M$ where $X$ is a topological space $X$ and $M$ is an abelian group. But let $X$ be the universal cover of a $K(G,1)$ space, in particular each $C_i(X,M)$ is $MG$ module for a ring $M$(say characterstic 0).
Question: Do we have a description of $C_i(X,M)$ in terms of $\mathbb{Z}G$-module $C_i(X,\mathbb{Z})$.
Doubt: If we take $C_i(X,\mathbb{Z})\otimes_{\mathbb{Z}G}M$, then we don't get a resolution as $Tor^{\mathbb{ZG}}(\mathbb{Z},M)$ would usually be non-zero.