Finite $2$-groups with exactly three automorphism orbits

Alexander Bors, Stephen P. Glasby

Published 2020-11-25Version 1

We give a complete classification of the finite $2$-groups $G$ such that the natural action of the automorphism group $\operatorname{Aut}(G)$ on $G$ admits exactly three orbits. By known results, this reduces the classification of all finite groups with exactly three automorphism orbits to the classification of those that are $p$-groups of exponent $p$ for some odd prime $p$.