Levi classes of quasivarieties of groups with commutator subgroup of order $p$

The Levi class generated by the class $\mathcal{M}$ of groups is the class of all groups in which the normal closure of each element belongs to $\mathcal{M}$. We describe Levi classes generated by a quasivariety $\mathcal{K}^{p^{s}}$ and some of its subquasivarieties, where $\mathcal{K}^{p^{s}}$ is the quasivariety of groups with commutator subgroup of order $p$ in which elements of the exponent of the degree of $p$ less than $p^{s}$ are contained in the center of the group, $p$ is a prime, $p\neq 2$, $s\geq 2$, and $s>2$ for $p=3$.