When computing the (co) homology of a group $G$, we can think about it in two related ways.
We define the homology of $G$ with coefficients in a $G$-module $M$ by constructing a resolution of $M$ over $\mathbb{Z}G$, removing $M$ from the resolution and tensoring the remaining terms with the $M$ (over $\mathbb{Z}G$ again). If you like, we have $H_n\left(G, M\right) = \text{Tor}^{\mathbb{Z}G}_n \left(\mathbb{Z}, M\right)$.
We define the homology of $G$ (but now only with $\mathbb{Z}$ coefficients) as the singular homology of the classifying space $BG$ (thinking of $G$ as a discrete topological group).
My question is as follows: In the first method, the homology is (strongly) dependent on the action of $G$ on $M$, however, in the second method we are restricted to considering $G$ with the trivial action on $\mathbb{Z}$. How do we generalise the topological approach to work with more interesting coefficient groups with possibly non-trivial actions of $G$?