Inner automorphisms act trivially on group cohomology

Let $k$ be a commutative ring, $G$ a group (both in the same universe), $M$ an object of $\mathrm{Rep}\,k\,G$, that is a $k$-linear representation $\rho = M.\rho$ of $G$, let $g \in G$ and let $n$ be a natural number. Write $c_g = (\mathrm{MulAut.conj}\ g)$ for the inner automorphism $h \mapsto g h g^{-1}$ of $G$, viewed as a monoid homomorphism, and let $\mathrm{Rep.res}\ c_g\ M$ be the representation with the same underlying $k$-module as $M$ and with $h$ acting by $\rho(g h g^{-1})$. Let $\varphi$ be a morphism of representations from $\mathrm{Rep.res}\ c_g\ M$ to $M$, and assume that on underlying modules $\varphi$ is given by $m \mapsto \rho(g^{-1}) m$. Then the map induced on cohomology by the pair $(c_g, \varphi)$ through the functoriality groupCohomology.map is the identity morphism of $H^n(G, M)$, for every degree $n$ — an equality of morphisms, not merely an equality after passing to cocycle classes.

This is the classical statement that inner automorphisms of $G$ act trivially on $H^n(G, M)$, in the form in which the pair (conjugation by $g$, multiplication by $\rho(g^{-1})$) induces the identity in every degree. It is used in the Herbrand-quotient part of the development, for instance when transporting classes along Shapiro-type isomorphisms and when comparing local fundamental classes and idelic cohomology classes under conjugation.