  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.
