On groups with given properties of the finite subgroups generated by couples of $2$-elements

Suppose that every finite subgroup, generated by a couple of $2$-elements of a periodic group, is either nilpotent of class 2 or of exponent 4. We prove that the group possesses the normal Sylow $2$-subgroup that is either nilpotent of class 2 or of exponent 4.