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, then the intersection $\text{Outc}(G)\cap \text{OutCol}(G)$ has odd order, where $\text{Outc}(G)$ and $\text{OutCol}(G)$ denote the class-preserving and Coleman outer automorphism groups, respectively. This implies that $G$ satisfies the normalizer problem for its integral group ring. Combined with earlier work on the dihedral and semidihedral cases, this settles the question for all three families of $2$-groups of $2$-rank two classified by Gorenstein–Walter and Alperin–Brauer–Gorenstein.