Generalized Periodicity in Group Cohomology

Given a finite group $G$, we introduce "encoding pairs," which are a pair of $G$-modules $M$ and $M'$ equipped with a shifted natural isomorphism between the cohomological functors $H^\bullet(G,\mathrm{Hom}_\mathbb Z(M,-))$ and $H^\bullet(G,\mathrm{Hom}_\mathbb Z(M',-))$. Studying these encoding pairs generalizes the theory of periodic cohomology for finite groups, allowing us to generalize the cohomological input of a theorem due to Swan that roughly says that a finite group with periodic cohomology acts feely on some sphere.