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**

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.)

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 $R$.

If $\sigma=\phi$ and $\sigma$ is not necessarily central, then $\Theta$ is an isomorphism iff $\sigma$ is central.

Thanks for any comments!