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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Aut</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \operatorname{Aut}(G) acting naturally on 𝐺 has three orbits. There are two infinite families and one additional group, of order <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mn>2</m:mn> <m:mn>9</m:mn> </m:msup> </m:math> 2^{9} . All of them are Suzuki 2-groups, and they appear (in a different context) in an earlier classification of Dornhoff.