  We study the nonabelian composition factors of a finite group $G$ assumed to
admit an $\operatorname{Aut}(G)$-orbit of length at least $\rho|G|$, for a
given $\rho\in\left(0,1\right]$. Our main results are the following: The orders
of the nonabelian composition factors of $G$ are then bounded in terms of
$\rho$, and if $\rho>\frac{18}{19}$, then $G$ is solvable. On the other hand,
for each nonabelian finite simple group $S$, there is a constant
$c(S)\in\left(0,1\right]$ such that $S$ occurs with arbitrarily large
multiplicity as a composition factor in some finite group $G$ having an
$\operatorname{Aut}(G)$-orbit of length at least $c(S)|G|$.
