Abelian automorphisms groups with a finite number of orbits

Let G be a group and A a group of automorphisms of G. An A-orbit of G is a set of the form {gα|α∈A}, where g is an element of G. The aim of this paper is to prove that if A is abelian and G is a union of a finite number of A-orbits then G admits a normal abelian subgroup of finite index. This result answers affirmatively a question raised by Neumann and Rowley (1998) in [4].