Let $K$ be a nonarchimedean local field, and $L/K$ be a finite Galois extension with Galois group $G={\rm Gal}(L/K)$ of order $n=[L:K]$. By local class field theory, there is a canonical isomorphism $$H^2(G,L^\times)\cong\frac{1}{n}{\Bbb Z}/{\Bbb Z}.$$ Let $u\in H^2(G,L^\times)$ be the fundamental class, that is, the generator corresponding to $\frac{1}{n}+{\Bbb Z}\in \frac{1}{n}{\Bbb Z}/{\Bbb Z}$.
Let $M$ be a torsion free $G$-module. For $r\in{\Bbb Z}$, consider the cup product map $$\alpha\colon H^r(G,M)\to H^{r+2}(G, M\otimes L^\times),\quad \xi \mapsto \xi\cup u.$$ By the Tate-Nakayama theorem, this map is an isomorphism.
Now let $L'\supset L\supset K$ be a larger finite Galois extension of $K$ with Galois group $G'={\rm Gal}(L'/K)$ of order $n'=[L':K]$. We have a cup product isomorphism $$\alpha'\colon H^r(G',M)\to H^{r+2}(G', M\otimes L^{\prime\,\times}),\quad \xi' \mapsto \xi'\cup u'$$ (where $u' \in H^2(G,L^{\prime\,\times})$ is the fundamental class), and an inflation map $$ {\rm Inf}\colon H^{r+2}(G, M\otimes L^\times)\to H^{r+2}(G', M\otimes L^{\prime\,\times}).$$
Question. What is the map marked with "?" making commutative the following diagram: $\require{AMScd}$ \begin{CD} H^r(G,M) @>\alpha>> H^{r+2}(G', M\otimes L^{\times})\\ @V?VV @VV{\rm Inf}V \\ H^r(G',M) @>\alpha'>> H^{r+2}(G', M\otimes L^{\prime\,\times}) \end{CD}
Note that ${\rm Inf}(u)=(n'/n)\cdot u'$.
I expect that the map $$ H^r(G,M) \overset?\longrightarrow H^r(G',M)$$ is just the inflation map induced by the surjective homomorphism $G'\to G$, but I cannot prove that.
