Let $G$ be a group and $H$ be a normal subgroup. For each $r \geq 0$, the inflation map from $H^r(G/H, M^H) \to H^r(G,M)$ is defined by the composition of two maps $H^r(G/H, M^H) \to H^r(G, M^H) \to H^r(G, M)$, where the first map is the restriction map $\text{res}$ and the second map is induced by the inclusion $M^H \to M$. I am wondering if this map can be seen as a map between two suitable $\delta$-functors. This is because most of the maps that I have encountered in group cohomology can be seen this way:
The restriction map $\text{res}$ can be seen as a $\delta$-functor map from $H^\bullet(G,-)$ to $H^\bullet(H, \rho^\sharp -)$, where $\rho^\sharp$ is the forgetful functor from $G$-Mod to $H$-Mod.
The transfer map $\text{tr}$ (when the index of $H$ in $G$ is finite) can be seen as a $\delta$-functor map from $H^\bullet(H, \rho^\sharp-)$ to $H^\bullet(G,-)$.
The isomorphism of Shapiro's lemma can be seen as a $\delta$-functor map between $H^\bullet (H, -)$ and $H^\bullet(G, \text{CoInd}_H^G -)$.
This makes them easy to work with because they commute nicely with connecting homomorphisms. I would like to know if the inflation map can also be put into the same framework.