Finite groups whose Sylow 2-subgroups have cyclic commutator subgroups

The following theorem is proved.
Theorem. {\it Suppose $G$ is a finite group such that $O^2(G)=G$ and $O_{2',2}(G)=O(G)$. Assume that a Sylow $2$-subgroup $T$ of $G$ is the direct product of subgroups $W$ and $A$, where $A$ is elementary Abelian and $W$ is non-Abelian dihedral, semidihedral, or wreathed. Then $T$ contains subgroups $W^*$ and $A^*$ with the following properties: $1)\ T=W^*\times A^*;$ $2)\ W\cong W^*,$ and all involutions of $W^*$ are conjugate in $G;$ $3)\ A\cong A^*,$ and $A^*$ is strongly closed in $T$ $($with respect to $G)$.}
As a consequence, a description is given of the finite groups whose Sylow 2-subgroups have cyclic commutator subgroups, the simple ones among which are the following: 1) $PSL_2(q)$, where $q\geqslant4$; 2) $PSL_3(q)$ and $PSU_3(q)$, where $q$ is odd; 3) $A_7$, $M_{11}$, the Janko group $J_1$, and the Ree groups.
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