Corestriction map in group cohomology (${\mathrm{Cor}}_U^G:H^q(U,M)\to H^q(G,M)$)

For an open finite-index subgroup $U$ of a profinite group $G$ and a discrete $G$-module $M$, cohomological transfer gives ${\mathrm{Cor}}_U^G:H^q(U,M)\to H^q(G,M)$. In degree zero it is the sum over cosets, or the norm for multiplicative coefficients. It is transitive and satisfies $\mathrm{Cor}\mathrm{Res}=[G:U]$ and the cup-product projection formula.