Relative Schur multipliers and universal extensions of group homomorphisms

In this note, starting with any group homomorphism $f\colon\Gamma\to G$, which is surjective upon abelianization, we construct a universal central extension $u\colon U\twoheadrightarrow G,$ UNDER $\Gamma$ with the same surjective property, such that for any central extension $m\colon M\twoheadrightarrow G,$ under $f,$ there is a unique homomorphism $U\to M$ with the obvious commutation condition. The kernel of $u$ is the relative Schur multiplier group $H_2(G,\Gamma;\mathbb{Z})$ as defined in the paper. The case where $G$ is perfect corresponds to $\Gamma=1$. This yields homological obstructions to lifting solution of equations in $G.$ Upon repetition, for finite groups, this gives a universal hypercentral factorization of the map $f\colon\Gamma\to G$.