A Levi class generated by a quasivariety of nilpotent groups

Let $L(M)$ be a class of all groups $G$ in which the normal closure of any element belongs to $M$; $qM$ is a quasivariety generated by a class $M$.
We consider a quasivariety $qH_2$ generated by a relatively free group in a class of nilpotent groups of class at most $2$ with commutator subgroup of exponent $2$. It is proved that the Levi class $L(qH_2)$ generated by the quasivariety $qH_2$ is contained in the variety of nilpotent groups of class at most $3$.