Commutator map is surjective homomorphism from exterior square to derived subgroup of central extension

Statement
Consider a short exact sequence of groups:

$$0 \to A \to E \to G \to 1$$

where $$E$$ is a central extension with central subgroup $$A$$ and quotient group $$G$$. Then, there is a natural homomorphism:

$$G \wedge G \to [E,E]$$

given by:

$$x \wedge y \mapsto [\tilde{x},\tilde{y}]$$

where $$\tilde{x},\tilde{y}$$ are elements of the extension group $$E$$ that map to $$x$$ and $$y$$ respectively.

Related facts

 * Commutator map is homomorphism from exterior square to derived subgroup
 * Hopf's formula for Schur multiplier
 * Commutator map on free group is isomorphism between exterior square and derived subgroup
 * Schur multiplier is kernel of commutator map homomorphism from exterior square to derived subgroup