Let $S$ and $R$ be groups and say $\sigma: S \twoheadrightarrow R$ is a group homomorphism that is a central extension; that is, it is surjective (extension) and its kernel is contained in the centre of $S$ (central). Let $\mathbf{ab} :\mathbf{Gp} \rightarrow\mathbf{Ab}$ be the abelianisation functor (the left adjoint to the inclusion of abelian groups into the category of groups). I would like to know of an elementary characterisation of the set of group homomorphisms $\phi: H \rightarrow R$ such that the canonical map $\Theta: S \times\_R H \rightarrow S \times\_{\mathbf{ab}S} \mathbf{ab}(S \times\_R H)$ is an isomorphism. 


**Notes for the question**


1. I am not assuming that $\phi$ is a surjection. If it were then Lemma 5.2.7 of 'Galois Theories' (Borceux and Janelidze) answers the question. (The answer is that $\Theta$ is iso. iff the commutator subgroups of $S$ and $S \times\_R H$ are isomorphic.)
2. I know that if $\Theta$ is iso. then both $\phi$ and $\pi\_2 : S \times\_R H \rightarrow H$ are central, but the converse is false. A counterexample is to take $\phi$ to be $e:1 \rightarrow R$ for some $R$ with non-trivial commutator subgroup where $e$ is the identity of $R$.
3. If $\sigma=\phi$ and $\sigma$ is not necessarily central, then $\Theta$ is an isomorphism iff $\sigma$ is central.


Thanks for any comments! 



Characterisation of a class of group homomorphisms related to a central extension
I don't have the book "Galois Theories" at hand but it seems to me that the result of the Lemma you mention should hold even if $\phi$ is not surjective.


Indeed, given a commutative diagram
$$\require{AMScd}\begin{CD}0 @>>> K @>{k}>> X @>{f}>> Y @>>> 0
\\ & @V{u}VV @V{v}VV @VV{w}V \\
0 @>>> K' @>>{k'}> X' @>>{f'}> Y' @>>> 0
\end{CD} \tag{1}\label{1}$$
where the rows are short exact sequences, then $u$ is an isomorphism if and only if the canonical arrow $X\to X'\times\_{Y'} Y$ is an isomorphism. Indeed, we have a diagram
$$\begin{CD}0 @>>> K @>{k}>> X @>{f}>> Y @>>> 0
\\ & @V{u}VV @V{(v,f)}VV @VV{1\_Y}V \\
0 @>>> K' @>{(k',0)}>> X'\times\_{Y'}Y @>{\psi\_2}>> Y @>>> 0
\\& @V{1\_{K'}}VV @V{\psi\_1}VV @VV{w}V
\\ 0 @>>> K' @>>{k'}> X' @>>{f'}> Y' @>>> 0,
\end{CD}$$
where the middle row is exact; hence we can apply the Short Five Lemma to the the two upper rows.


Then it suffices to apply this to the case where the short exact sequences in \eqref{1} are given by the abelianizations of $S\times\_HR$ and $S$.


