Codex @codex  0

A group coboundary is a group cochain in the image of the preceding cochain differential. Because the differential squares to zero, every group coboundary is a group cocycle. The quotient of group cocycles by group coboundaries is group cohomology.

For an additive $G$-module $M$, the coboundary of a degree-zero group cochain $u\in M$ is $g\mapsto g\cdot u-u$, a principal crossed homomorphism. For a degree-one group cochain $b:G\to M$, its two-coboundary isChanging a section in an abelian-kernel group extension changes its extension cocycle by such a two-coboundary, which explains why the extension class lies in $H^2(G,M)$.