Sylow $2$-Subgroups of Finite Groups with a Single Conjugacy Class of Involutions

We solve Brauer’s Problem 33 by determining the finite $2$-groups that occur as Sylow $2$-subgroups of finite groups with a single conjugacy class of involutions. Besides the Sylow $2$-subgroups arising from the relevant non-abelian simple groups in Table 1, these are precisely the homocyclic, generalized quaternion, and Suzuki $2$-groups. The classification of finite simple groups is used in this paper.