Functorality of universal central extension

Question

I'm looking a reference (or quick proof) for the following fact, which doesn't appear in the standard sources I've consulted (for instance, Milnor's book on algebraic k-theory).

Let $\phi : G \rightarrow G'$ be a homomorphism between perfect groups. Set $A = H_2(G)$ and $A' = H_2(G')$, and let $$1 \longrightarrow A \longrightarrow \tilde{G} \longrightarrow G \longrightarrow 1$$ and $$1 \longrightarrow A' \longrightarrow \tilde{G}' \longrightarrow G' \longrightarrow 1$$ be the universal central extensions. Then there exists a unique homomorphism $\tilde{\phi} : \tilde{G} \rightarrow \tilde{G}'$ lifting $\phi$. Also, the restriction $A \rightarrow A'$ of $\tilde{\phi}$ to $A$ is the map $A \rightarrow A'$ induced by $\phi$.

Answer 1

Two sources come to mind, though there may be more recent ones. These are both available online now.

1) A concise exposition of central extensions is given by Steinberg in Section 7 of his 1967-68 Yale lectures on Chevalley groups, distributed in mimeographed form and currently linked on his UCLA homepage here. Item (ix) at the bottom of page 76 is what you need. The proof is elementary, and standard, but note an obvious typo on page 77: the first handwritten symbol $\psi$ should have been $\pi$.

Originally Steinberg developed these ideas mainly in order to justify lifting of projective representations of finite Chevalley groups to ordinary linear representations of such groups of "universal" type. But along the way he worked out in considerable generality the algebraic analogues of topological connectedness and simple connectedness: the former imitated by a group being perfect (equal to its derived group) and the latter by a group being a universal central extension of a given group (possible only when the given group is perfect).

2) An even more concise account can be found in the important 1968 paper by Calvin Moore here. See his statement in the middle of page 10. As indicated by the title Group extensions of $p$-adic and adelic linear groups, Moore came to these ideas from a rather different direction. But his approach and Steinberg's soon merged in the work of Matsumoto and others on the Congruence Subgroup Property in the case of Chevalley (split) groups over number fields.

P.S. Concerning your last line, the formalism for the algebraic "fundamental group" varies (homology vs. cohomology), but I think the treatments of central extensions given by Steinberg and others yield immediately what you want. Work on the CSP tends to emphasize the cocycle approach here, featuring Steinberg cocycles in the special cases of interest. (There is some exposition with references in the last part of my 1980 Arithmetic Groups, Springer Lecture Notes 789.)