Levi quasivarieties of exponent $p^s$

For an arbitrary class $M$ of groups, $L(M)$ denotes a class of all groups $G$ the normal closure of any element in which belongs to $M$; $qM$ is a quasivariety generated by $M$. Fix a prime $p$, $p\ne2$, and a natural number $s$, $s\ge2$. Let $qF$ be a quasivariety generated by a relatively free group in a class of nilpotent groups of class at most 2 and exponent $p^s$, with commutator subgroups of exponent $p$. We give a description of a Levi class generated by $qF$.
Fix a natural number $n$, $n\ge2$. Let $K$ be an arbitrary class of nilpotent groups of class at most $2$ and exponent $2^n$, with commutator subgroups of exponent $2$. Assume also that for all groups in $K$, elements of order $2^m$, $0<m<n$, are contained in the center of a given group. It is proved that a Levi class generated by a quasivariety $qK$ coincides with a variety of nilpotent groups of class at most $2$ and exponent $2^n$, with commutator subgroups of exponent $2$.