  Let $G$ be a group. The orbits of the natural action of $\mbox{Aut}(G)$ 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 $\omega(G) \leq 6$ 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 $\omega(G) \leq n$,
generalizing a result of Kohl.
