So let $X$ be a "nice" topological space and assume that $G$ is a finite group which acts freely on $X$.
Q: Is there a simple relationship between the cohomology groups $H^i(G,\mathbf{Z}), H^i(X,\mathbf{Z})$ and $H^i(X/G,\mathbf{Z})$? Does the Leray spectral sequence simplifies in this special case?