Change of group commutes with the connecting homomorphism

Let $k$ be a commutative ring, $G$ and $G'$ groups and $\pi\colon G'\to G$ a group homomorphism. Let $X$ be a short complex $X_1\xrightarrow{f}X_2\xrightarrow{g}X_3$ in $\mathrm{Rep}\,k\,G$ which is short exact, and $X'$ a short exact short complex $X'_1\to X'_2\to X'_3$ in $\mathrm{Rep}\,k\,G'$. Let $\varphi_1,\varphi_2,\varphi_3$ be morphisms of $k$-linear $G'$-representations $\mathrm{Rep.res}\,\pi\,X_i\to X'_i$, that is, from the restriction of $X_i$ along $\pi$ to $X'_i$, and assume the two squares commute: the restriction of $f$ followed by $\varphi_2$ equals $\varphi_1$ followed by $X'.f$, and the restriction of $g$ followed by $\varphi_3$ equals $\varphi_2$ followed by $X'.g$. Let $i,j$ be natural numbers with $i+1=j$ and let $y$ be an element of $H^i(G,X_3)$. Then the image of $\delta_X(y)\in H^j(G,X_1)$ under the change-of-group map $H^j(G,X_1)\to H^j(G',X'_1)$ attached to $\pi$ and $\varphi_1$ equals the image under the connecting map $\delta_{X'}$ of the element of $H^i(G',X'_3)$ obtained from $y$ by the change-of-group map attached to $\pi$ and $\varphi_3$.

This is the naturality of the connecting homomorphism in the long exact cohomology sequence with respect to change of group, restriction along $\pi$ followed by push-forward along the $\varphi_i$ (inflation when $\pi$ is a quotient map, and naturality in the short exact sequence when $\pi$ is the identity), stated in the element-wise form in which it is used. It serves to transport the long exact sequence attached to a short exact sequence of relation modules along a tower of groups.