Virtually nilpotent groups with finitely many orbits under automorphisms

Let G be a group. The orbits of the natural action \({{\,\mathrm{Aut}\,}}(G)\) on are called automorphism G, and number is denoted by \(\omega (G)\). virtually nilpotent group such that (G)< \infty \). We prove \(G = K \rtimes H\) where H torsion subgroup torsion-free radicable characteristic G. Moreover, we \(G^{'}= D \times {{\,\mathrm{Tor}\,}}(G^{'})\) subgroup. In particular, if maximum normal \(\tau (G)\) trivial, then \(G^{'}\) nilpotent.