Many thanks, Derek Holt! A quick observation: in the case where $K \subset [\overline{H},\overline{G}]$ (this happens for instance when $\operatorname{Cor}:\operatorname{H}_2(H,\mathbb{Z}) \to \operatorname{H}_2(G,\mathbb{Z})$ is surjective) the resulting quotient is also isomorphic to $\frac{H \cap [G,G]}{[H,G]}$.

@AndreiSmolensky, many thanks, but why is $H_2(\overline{G},\mathbb{Z})=0$? For $G=V_4$ we can have $\overline{G} \cong D_4$, which does not have trivial Schur multiplier...