  In this paper we study virtual rational Betti numbers of a
nilpotent-by-abelian group $G$, where the abelianization $N/N'$ of its
nilpotent part $N$ satisfies certain tameness property. More precisely, we
prove that if $N/N'$ is $2(c(n-1)-1)$-tame as a $G/N$-module, $c$ the
nilpotency class of $N$, then
$\mathrm{vb}_j(G):=\sup_{M\in\mathcal{A}_G}\dim_\mathbb{Q} H_j(M,\mathbb{Q})$
is finite for all $0\leq j\leq n$, where $\mathcal{A}_G$ is the set of all
finite index subgroups of $G$.
