I know if $k$ is a field such that $char(k)$ divides $|G|$ (a finite group), then finding the ring structure on $H^\ast(G,k)$ can be very, very hard.
But what about when $k=\mathbb{Z}$? Is the computation easier? I know of several "modern" techniques to compute the cohomology groups, but have never encountered a detailed computation of the ring structure.
Thanks!