Relative Schur multipliers and universal extensions of group homomorphisms

We discuss an extension to a relative case of the construction by Schur of the universal central extension of a perfect group G, with the kernel being the Schur multiplier group H₂(G; ℤ). In this note, starting with any group homomorphism f : Γ → G, which is surjective upon abelianization, we construct a universal central extension u: U ↠ G, relative to f with the same surjective property, such that for any central extension m: M ↠ G, relative to f, there is a unique homomorphism U → M with the obvious commutation condition. The kernel of u is the relative Schur multiplier group H₂(G, Γ Z) as below. The case where G is perfect corresponds to Γ = 1. Upon repetition, for finite groups, this also gives a universal hypercentral factorization of the map f : Γ → G. We observe that our construction of those relative universal central extensions yield homological obstructions to lifting solutions of equations in a perfect group G to its Schur universal central extension E ↠ G.