Let $G$ be a finite group and $A$ a $G$-Module. It is well-known that $H^q(G,A)$ is killed by $|G|$ for all $q \geq 1$. This is usually proved using Restriction-Corestriction (applied with the trivial subgroup).
Is there a way to prove this for $q=1$ without using this machinery? Specifically, is there a computation one can do at the level of cocycles which gives this result for $H^1(G,A)$?