The Levi classes generated by nilpotent groups

Given an arbitrary class $M$ of groups, denote by $L(M)$ the class of all groups $G$ in which the normal closure of every element belongs to $M$. Consider the quasivariety $qF_p$ generated by the relatively free group in the class of nilpotent groups of length at most 2 with the commutant of exponent $p$ (where $p$ is an odd prime). We describe the Levi class that is generated by $qF_p$.