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Abstract We give a complete classification of the finite 2-groups 𝐺 for which the automorphism group Aut (G) Aut (G) acting naturally on 𝐺 has three orbits. There are two infinite families and one additional group, of order 2 9 2^9. All of them are Suzuki 2-groups, and they appear (in a different context) in an earlier classification of Dornhoff.
Bors et al. (Wed,) studied this question.
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