Second group cohomology classifies group extensions

Equivalence classes of group extensions of $G$ by an abelian $G$-module $A$ correspond to $H^2(G,A)$. A section $s:G\to E$ produces the extension cocycleand changing the section changes $c$ by a group coboundary.

An extension cocycle is the two-cocycle obtained from a section of a group extension. It measures the failure of the section to be a group homomorphism.