Finite $2$-groups with exactly three automorphism orbits

TL;DR

This paper classifies finite 2-groups with exactly three automorphism orbits, identifying two infinite families and one specific group, all of which are Suzuki 2-groups, expanding understanding of automorphism group actions.

Contribution

It provides a complete classification of finite 2-groups with three automorphism orbits, including explicit families and a unique example, linking to Suzuki 2-groups and prior classifications.

Findings

Two infinite families of such groups identified

One unique group of order 2^9 classified

All groups are Suzuki 2-groups

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.