Finite $2$-groups with exactly three automorphism orbits

Abstract

We give a complete classification of the finite $2$-groups $G$ for which the automorphism group $\operatorname{Aut}(G)$ acting naturally on $G$ has three orbits. There are two infinite families and one additional group, of order $2^9$. All of them are Suzuki $2$-groups, and they appear in an earlier classification of Dornhoff.