The axiomatic rank of Levi classes

A Levi class $L(\mathcal M)$ generated by a class $\mathcal M$ of groups is a class of all groups in which the normal closure of each element belongs to $\mathcal M$.
It is stated that there exist finite groups $G$ such that a Levi class $L(qG)$, where $qG$ is a quasivariety generated by a group $G$, has infinite axiomatic rank. This is a solution for [The Kourovka Notebook, Quest. 15.36].
Moreover, it is proved that a Levi class $L(\mathcal M)$, where $\mathcal M$ is a quasivariety generated by a relatively free $2$-step nilpotent group of exponent ps with a commutator subgroup of order $p$, $p$ is a prime, $p\ne2$, $s\ge2$, is finitely axiomatizable.