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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.

A group extension is split when its quotient homomorphism admits a group-homomorphic section. Equivalently, the middle group is a semidirect product of the kernel by the quotient.

A group is virtually solvable, or virtually soluble, when it contains a solvable group as a finite-index subgroup. This is preserved under subgroups, quotients and group extensions. The extension proof can use a finite-index characteristic soluble subgroup of the kernel, followed by centralizing the resulting finite normal kernel. Finite generation is not required for these closure properties.