  If $G$ is a nilpotent group with a balanced presentation and
$G\not\cong\mathbb{Z}^3$ then $\beta_1(G;\mathbb{Q})\leq2$ \cite{Hi22}. We show
that if such a group $G$ has an abelian normal subgroup $A$ such that
$G/A\cong\mathbb{Z}^2$ then $G$ is torsion-free and has Hirsch length
$h(G)\leq4$. On the other hand, if $\beta_1(G;\mathbb{Q})=1$ and $G$ has an
abelian normal subgroup $A$ such that $G/A\cong\mathbb{Z}$ then
$G\cong\mathbb{Z}/m\mathbb{Z}\rtimes_n\mathbb{Z}$, for some $m,n\not=0$ such
that $m$ divides a power of $n-1$.
