Relative Schur multipliers and universal extensions of group homomorphisms

In this note, starting with any group homomorphism $f\text{\hspace{0.17em}⁣}:\mathrm{\Gamma }\to G$, which is surjective upon abelianization, we construct a universal central extension $u\text{\hspace{0.17em}⁣}:U\twoheadrightarrow G,$ UNDER $\mathrm{\Gamma }$ with the same surjective property, such that for any central extension $m\colon M\twoheadrightarrow G,$under$f,$thereisauniquehomomorphism$U\to M$ with the obvious commutation condition. The kernel of $u$ is the relative Schur multiplier group $H_2(G,\mathrm{\Gamma };\mathbb{Z})$ as defined in the paper. The case where $G$ is perfect corresponds to $\mathrm{\Gamma }=1$. This yields homological obstructions to lifting solution of equations in $G\mathrm{.}$ Upon repetition, for finite groups, this gives a universal hypercentral factorization of the map $f\text{\hspace{0.17em}⁣}:\mathrm{\Gamma }\to G$.