  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 $\omega(G)$. Let $G$ be a virtually nilpotent group such that
$\omega(G)< \infty$. We prove that $G = K \rtimes H$ where $H$ is a torsion
subgroup and $K$ is a torsion-free nilpotent radicable characteristic subgroup
of $G$. Moreover, we prove that $G^{'}= D \times \Tor(G^{'})$ where $D$ is a
torsion-free nilpotent radicable characteristic subgroup. In particular, if the
maximum normal torsion subgroup $\tau(G)$ of $G$ is trivial, then $G^{'}$ is
nilpotent.
