Periodic groups, saturated by wreathed groups

It is proved that an infinite 2–group saturated by the set $\mathfrak{S}=\{(<a>\times <b>)\leftthreetimes(v)| \ |a|=|b|=2^n, v^2=e, a^v=b, n=1,2,...\}$ is isomorphic to the wreath product of a locally cyclic group and a group of order 2.