Nilpotent groups with balanced presentations. II

If $G$ is a nilpotent group with a balanced presentation and $G\cong ̸{\mathbb{Z}}^3$ then ${\beta }_1(G;\mathbb{Q})\le 2$ \cite{Hi22}. We show that if such a group $G$ has an abelian normal subgroup $A$ such that $G\mathrm{/}A\cong {\mathbb{Z}}^2$ then $G$ is torsion-free and has Hirsch length $h(G)\le 4$. On the other hand, if ${\beta }_1(G;\mathbb{Q})=1$ and $G$ has an abelian normal subgroup $A$ such that $G\mathrm{/}A\cong \mathbb{Z}$ then $G\cong \mathbb{Z}\mathrm{/}m\mathbb{Z}{⋊}_n\mathbb{Z}$, for some $m,n≠0$ such that $m$ divides a power of $n-1$.