Class-preserving Coleman automorphisms of finite groups with Wreathed Sylow 2-subgroups

We show that if G is a finite group whose Sylow 2-subgroups are wreathed 2-groups W (C2ⁿ × C2ⁿ) C₂ W ≅ ( C 2 n × C 2 n ) ⋊ C 2 with n ≥ 2 n ≥ 2 , then the intersection Outc(G) ∩ OutCol(G) Out c ( G ) ∩ Out Col ( G ) has odd order, where Outc(G) Out c ( G ) and OutCol(G) Out 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 ( PSL(2,q) PSL ( 2 , q ) with q odd, q ≥ 5 q ≥ 5 , and the alternating group A₇ A 7 ) and by Alperin, Brauer, and Gorenstein in the semidihedral and wreathed cases (the groups PSL(3,q) PSL ( 3 , q ) and PSU(3,q) PSU ( 3 , q ) for suitable odd q , with the Mathieu group <jats:alternative