Wreath products in modular group algebras of some finite 2-groups

Let $K$ be field of characteristic 2 and let $G$ be a finite non-abelian 2-group with the cyclic derived subgroup $G^{\mathrm{\prime }}$, and there exists a central element $z$ of order 2 in $Z(G)\mathrm{\backslash }{G}^{\mathrm{\prime }}$. We prove that the unit group of the group algebra $KG$ possesses a section isomorphic to the wreath product of a group of order 2 with the derived subgroup of the group $G$, giving for such groups a positive answer to the question of A. Shalev.