Homology of free Abelianized extensions of groups

Let $G$ be a group given by a free presentation $G=F/N$, and $N'$ the commutator subgroup of $N$. The quotient $F/N'$ is called a free abelianized extension of $G$. We study the homology of $F/N'$ with trivial coefficients. In particular, for torsion-free $G$ our main result yields a complete description of the odd torsion in the integral homology of $F/N'$ in terms of the mod $p$ homology of $G$.