If one knows the commutator subgroup $[G, G]$ and the abelianization $G/[G, G]$ of a group $G$, what else is needed to determine the group? If $G$ is a central extension of a group $H$, $G$ is determined by $Z(G)$, $H$ and a cocycle. Is there a corresponding theory for extensions with a given commutator subgroup?
For example, is it possible to define a group operation on $[G, G] \times G/[G, G]$ so that the resulting group is isomorphic to $G$, as it happens for central extensions?
It is natural that extensions $E$ fit a sequence $$0\to [G,G]\to E\to G/[G,G]\to0,$$ and these $E$ are classified by homomorphisms $$G/[G,G]\to {\rm Out}[G,G]$$ where ${\rm Out}[G,G]=\dfrac{{\rm Aut}[G,G]}{{\rm Inn}[G,G]}$.