 Periodic group cohomology

For a finite group, the usual integral notion asks for a class $\Delta \in H^n(G;\mathbb{Z})$ whose cup product gives the displayed isomorphisms for every coefficient module $M$ and every $i>0$. The analogous claim for just trivial coefficients over a fixed field is weaker and should be stated as such. A free cohomologically orientation-preserving sphere action supplies that field-coefficient version through periodic group cohomology from a free sphere action.