The quasivariety generated by a torsion-free Abelian-by-finite group

Let $L_q(qG)$ be the quasivariety lattice contained in a quasivariety generated by a group $G$. It is proved that if $G$ is a finitely generated torsion-free group in $\mathcal A\mathcal B_{2^n}$ (i.e., $G$ is an extension of an Abelian group by a group of exponent $2^n$), which is a split extension of an Abelian group by a cyclic group, then the lattice $L_q(qG)$ is a finite chain.