  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 $\omega(G)$. We prove that if $G$ is a soluble group with
finite rank such that $\omega(G)< \infty$, then $G$ contains a torsion-free
characteristic nilpotent subgroup $K$ such that $G = K \rtimes H$, where $H$ is
a finite group. Moreover, we classify the mixed order soluble groups of finite
rank such that $\omega(G)=3$.
