We show that if G is a finite group whose Sylow 2-subgroups are wreathed 2-groups \(W \cong (C_{2^n} \times C_{2^n}) \rtimes C_2\) with \(n \ge 2\) , then the intersection \(\operatorname {Out}_c(G) \cap \operatorname {Out}_{\textrm{Col}}(G)\) has odd order, where \(\operatorname {Out}_c(G)\) and \(\operatorname {Out}_{\textrm{Col}}(G)\) denote the class-preserving and Coleman outer automorphism groups, respectively. In particular, G satisfies the normalizer condition for its integral group ring. Together with earlier results for the dihedral and semidihedral cases, this settles the question for all finite groups whose Sylow 2-subgroups are of 2-rank two. We recall that the finite simple groups with such Sylow 2-subgroups were classified by Gorenstein and Walter in the dihedral case ( \(\textrm{PSL}(2,q)\) with q odd, \(q \ge 5\) , and the alternating group \(A_7\) ) and by Alperin, Brauer, and Gorenstein in the semidihedral and wreathed cases (the groups \(\textrm{PSL}(3,q)\) and \(\textrm{PSU}(3,q)\) for suitable odd q, with the Mathieu group \(M_{11}\) appearing in the semidihedral case only).
