  We study finite groups $G$ such that the maximum length of an orbit of the
natural action of the automorphism group $\operatorname{Aut}(G)$ on $G$ is
bounded from above by a constant. Our main results are the following: Firstly,
a finite group $G$ only admits $\operatorname{Aut}(G)$-orbits of length at most
$3$ if and only if $G$ is cyclic of one of the orders $1$, $2$, $3$, $4$ or
$6$, or $G$ is the Klein four group or the symmetric group of degree $3$.
Secondly, there are infinitely many finite ($2$-)groups $G$ such that the
maximum length of an $\operatorname{Aut}(G)$-orbit on $G$ is $8$. Thirdly, the
order of a $d$-generated finite group $G$ such that $G$ only admits
$\operatorname{Aut}(G)$-orbits of length at most $c$ is explicitly bounded from
above in terms of $c$ and $d$. Fourthly, a finite group $G$ such that all
$\operatorname{Aut}(G)$-orbits on $G$ are of length at most $23$ is solvable.
