Levi classes of quasivarieties of nilpotent groups of class at most two

A Levi class $L(\mathcal{M})$ generated by a class $\mathcal{M}$ of groups is the class of all groups in which the normal closure of every cyclic subgroup belongs to $\mathcal{M}$. Let $p$ be a prime and $p\neq 2$, let $H_{p}$ be a free group of rank $2$ in the variety of nilpotent groups of class at most $2$ with commutator subgroup of exponent $p$, and let $qH_{p}$ be the quasivariety generated by the group $H_{p}$. It is shown that there exists a set of quasivarieties $\mathcal{M}$ of cardinality continuum such that $L(\mathcal{M})=L(qH_{p})$. Let $s$ be a natural number, $s\geq 2$. We specify a system of quasi-identities defining $L(q(H_{p}, Z_{p^{s}}))$, and prove that there exists a set of quasivarieties $\mathcal{M}$ of cardinality continuum such that $L(\mathcal{M})=L(q(H_{p}, Z_{p^{s}}))$, where $Z_{p^{s}}$ is a cyclic group of order $p^{s}$; $q(H_{p}, Z_{p^{s}})$ is the quasivariety generated by the groups $H_{p}$ and $Z_{p^{s}}$.