In this paper, a further study of the Levi classes is presented. The Levi class generated by the variety of exponent 2ρ groups with the commutant exponent ρ and commuting squares of elements (ρ is a prime, ρ ≠ 2) are described. A cover for a group G is a collection of G subgroups which set-theoretic union is G. A study of the influence of the cover properties on the group structure is one of the important directions of a group theory. The study of group G properties inherited from the groups properties of a certain cover of group G is of special interest. For an arbitrary class of groups M a class of all groups G where the normal closure of every element of G belongs to M is denoted by L(M). The class L(M) is called the Levi class generated by M. Study of Levi classes should be considered as a step towards studying the structure of the groups covered by a system of normal subgroups. Levi classes were introduced under the influence of Levi’s article where Levi gave the classification of groups with abelian normal closures. R.F. Morse proved that if the class M is a variety of groups then L(M) is also a variety of groups. According to research of A.I. Budkin it follows that if M is a quasivariety then L(M) is also a quasivariety of groups. Earlier, author found the descriptions of Levi classes generated by the almost Abelian quasivarieties of nilpotent groups (except one).