I'm learning about homology of groups right now and I have a general question about homology of subgroups. I know that we won't get "homology of subgroup is subgroup of homology", that wouldn't make sense for several reasons. But, I am trying to figure out what can be said. If $J\leq G$ is a subgroup and $X$ is a $K(G,1)$, then by Galois correspondence we have a covering space $p:Y\to X$ such that $\pi_1(Y)\cong J$. Moreover, since $X$ and $Y$ have the same universal cover, $Y$ is a $K(J,1)$. So the homology of $J$ is by definition the homology of $Y$. We know that $H_*(\cdot)$ on topological spaces is a functor, so we have a map on homology $p_*:H_*(Y)\to H_*(X)$. But what does $p$ do exactly?
EDIT: A quick example. Consider $G=\langle x| \ \rangle$. A $K(G,1)$ is the unit circle, which we may assume is embedded in $\mathbb{C}$. Then the subgroup $J=\langle 2x \rangle$ corresponds to the covering $p:S^1\to S^1$ by $z^2$. I am not sure what $p_*$ does, but I already know that $H_*(J)\cong H_*(G)$. So it seems like in order to start talking about this general problem we need to start adding constraints: like choose $J\leq G$ with $J\ncong G$.