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Let  G  be a group. The orbits of the natural action ofAut  ⁡   (  G  ){\operatorname{Aut}(G)}    on  G  are called “automorphism orbits” of  G , and the number of automorphism orbits of  G  is denoted byω  ⁢   (  G  ){\omega(G)}   . In this paper the finite non-solvable groups  G  withω  ⁢   (  G  )    ≤  6{\omega(G)\leq 6}    are classified – this solves a problem posed by Markus Stroppel – and it is proved that there are infinitely many finite non-solvable groups  G  withω  ⁢   (  G  )    =  7{\omega(G)=7}   . Moreover, it is proved that for a given number  n  there are only finitely many finite groups  G  without non-trivial abelian normal subgroups and such thatω  ⁢   (  G  )    ≤  n{\omega(G)\leq n}   , generalizing a result of Kohl.

Finite groups with six or seven automorphism orbits