= Group coboundary
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> is
$$
(\delta b)(g,h)=g\cdot b(h)-b(gh)+b(g).
$$
Changing 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 $H^2(G,M)$.
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