Two-coboundary ($\delta b(g,h)=g\cdot b(h)-b(gh)+b(g)$)

For an additive left $G$-module $M$, a two-coboundary is a degree-two group cochain obtained from a degree-one cochain $b:G\to M$ byIt is automatically a two-cocycle because ${\delta }^2=0$. Expanding the four terms in $g\cdot (\delta b)(h,k)-(\delta b)(gh,k)+(\delta b)(g,hk)-(\delta b)(g,h)$ makes every contribution cancel.

For trivial action and $G=\mathbb{Z}$, take $M=\mathbb{Z}$ and $b(n)=n^2$. Then $(\delta b)(m,n)=n^2-(m+n{)}^2+m^2=-2mn$, a nonzero cochain representing the zero class in $H^2(G,M)$. Being cohomologically trivial does not require being pointwise zero.

Changing a section in an abelian-kernel group extension changes its extension cocycle by a two-coboundary. This is the mechanism behind quotienting two-cocycles by two-coboundaries when defining second group cohomology.