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Let $G$ be a group. The orbits of the natural action of Aut$(G)$ on $G$ arecalled ``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 withfinite rank such that $\omega(G)< \infty$, then $G$ contains a torsion-freecharacteristic nilpotent subgroup $K$ such that $G = K \rtimes H$, where $H$ isa finite group. Moreover, we classify the mixed order soluble groups of finiterank such that $\omega(G)=3$.

Soluble Groups with few orbits under automorphisms