Virtually nilpotent groups with finitely many orbits under automorphisms

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) < infinity. We prove that G = K (sic) 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 x 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.