Finite Groups with 6 or 7 Automorphism Orbits

Let $G$ be a group. The orbits of the natural action of $Aut(G)$ on $G$ are called “automorphism orbits” of $G$, and the number of automorphism orbits of $G$ is denoted by $\omega (G)$. In this paper the finite nonsolvable groups $G$ with $\omega (G)\le 6$ are classified - this solves a problem posed by Markus Stroppel - and it is proved that there are infinitely many finite nonsolvable groups $G$ with $\omega (G)=7$. Moreover it is proved that for a given number $n$ there are only finitely many finite groups $G$ without nontrivial abelian normal subgroups and such that $\omega (G)\le n$, generalizing a result of Kohl.