On page 338, A User's Guide to Spectral Sequences. 2nd Edition, by John McCleary, Theorem 8.9, there is a Cartan-Leray spectral sequence for homology:

If $X$ is a connected pace on which the group $\pi$ acts freely and properly, then there is a spectral sequence, homological type, with $$ E^2_{p,q}=H_p(\pi,H_q(X)),$$ converging strongly to $H_*(X/\pi)$.

Is there any cohomology version of Cartan-Leray spectral sequence that can get cup product structure of $H^*(X/\pi)$?

On page 338, A User's Guide to Spectral Sequences. 2nd Edition, by John McCleary, Theorem 8.9, there is a Cartan-Leray spectral sequence for homology:

If $X$ is a connected pace on which the group $\pi$ acts freely and properly, then there is a spectral sequence, homological type, with $$ E^2_{p,q}=H_p(\pi,H_q(X)),$$ converging strongly to $H_*(X/\pi)$.

Is there any cohomology version of Cartan-Leray spectral sequence that can get cup product structure of $H^*(X/\pi)$?

Let $k$ be a field. Suppose $H^*(X;k)$ and $H^*(\pi;k)$ are known. How to get the cup product structure of $H^*(X/\pi;k)$?