On the quasivarieties generated by a finite group and lacking any independent bases of quasi-identities

Let ${\Cal R}_{p^k}$ be the variety of $2$-nilpotent groups of exponent $p^k$ with commutator subgroup of exponent $p$ ($p$ is a prime). We prove the infinity of the set of the subquasivarieties of ${\Cal R}_{p^k}$ $(k\geq 2)$ generated by a finite group and lacking any independent bases of quasi-identities.