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 $W \cong (C_{2^n} \times C_{2^n}) \rtimes C_2$ with $n \geq 2$, then the intersection $\Outc(G) \cap \OutCol(G)$ has odd order, where $\Outc(G)$ and $\OutCol(G)$ 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 ($\PSL(2,q)$ with $q$ odd, $q \geq 5$, and the alternating group $A_7$) and by Alperin, Brauer, and Gorenstein in the semidihedral and wreathed cases (the groups $\PSL(3,q)$ and $\PSU(3,q)$ for suitable odd~$q$, with the Mathieu group $M_{11}$ appearing in the semidihedral case only).