Restriction map on cohomology and automorphisms of $G$

Question

I'm trying to prove the following lemma on group cohomology, which seems obvious to me when I state it, but I'm not finding a satisfactory way to write down the demonstration.

Let $H\leq G$ be a subgroup of a finite group, and let $A$ be a $G$-module. If $z \in H^n(H, A)$ satisfies $z = \text{res}^G_H w$ for some $w \in H^n(G, A)$, then $gz = \text{res}^G_{gHg^{-1}} w$ for all $g \in G$.

Here, $\text{res}^G_H\colon H^n(G, A) \to H^n(H, A)$ is the usual restriction homomorphism induced by the inclusion $H \subseteq G$, and the map $g(-)\colon H^n(H, A) \to H^n(gHg^{-1}, A)$ is induced by the isomorphism $ghg^{-1} \mapsto h$ and the module homomorphism $a \mapsto ga$.

After fiddling around, I found that for all $w \in H^n(G, A)$ it is sufficient to prove that $\text{res}^G_{gHg^{-1}} \varphi_g(w) = \text{res}^G_{gHg^{-1}} w$, where $\varphi_g$ is the automorphism of $H^n(G, A)$ induced by conjugation by $g^{-1}$ and translation by $g$ of the module $A$. I'm not sure how to go about proving this though.

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What the lemma essentially states is, given a function $z$ who is a restriction of a function $w$ to the domain $H$, the function $gz$ obtained by conjugating $H$ and then applying $z$ is equal to restricting $w$ to the conjugate domain $gHg^{-1}$. This seems obvious to me, so obvious that I'm not sure how to make it rigorous.

Answer 1

After some searching, I found out that this is an easy consequence of the following proposition:

Let $g$ be a fixed element of a group $G$, and consider the maps $f\colon G \to G$ and $j\colon A \to A$ for an arbitrary module $A$ given by $f(x) = gxg^{-1}$ and $j(a) = g^{-1}a$. Then, the induced isomorphism $$(f, j)\colon H^n(G, A) \to H^n(G, A)$$ is the identity.

This is proven, for instante, in Hilton & Stammbach's A Course in Homological Algebra, Chapter VI, Proposition 16.2. The idea of the proof is rather simple, embedding $A$ in an injective module $I$, and using induction on $n$ looking at the long exact sequence induced by $$\{0\} \to A \to I \to I/A \to \{0\}$$