On the orders of automorphism groups of finite groups

In the Kourovka notebook, Deaconescu asks whether vertical bar AutG vertical bar >= phi(vertical bar G vertical bar) for all finite groups G, where phi denotes the Euler totient function, and whether G is cyclic whenever vertical bar AutG vertical bar = phi(vertical bar G vertical bar). Both questions are answered in the negative in this paper. Moreover, vertical bar AutG vertical bar/phi(vertical bar G vertical bar) can be made arbitrarily small.