Finite-group cohomology is annihilated by the group order

If $G$ is finite and $n>0$, multiplication by $|G|$ annihilates $H^n(G,M)$. Indeed, restriction to the trivial subgroup is zero in positive degree, while the restriction-corestriction composite is multiplication by $|G|$.

For a subgroup $K\le G$, restriction followed by corestriction acts on group cohomology as the sum over coset translates. For $K$ trivial, inner automorphisms act trivially on cohomology, so this sum is multiplication by $|G|$.