  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.
