Let $f:H\hookrightarrow G$ and $g:M\to N$ be morphisms of groups ($f$ is injection) and modules (respectively), where $M,N\in\text{Mod}(G)$. By the functoriality of the cohomology we get induced maps by $f$ and $g$ on the cohomology, and I'll put all the information above in a diagram, and my question is: Is this diagram commutative?
Thanks in advance.