In the Kourovka Notebook, Deaconescu asks if |Aut G| ≥ φ(|G|) for all finite groups G, where φ denotes the Euler totient function; and whether G is cyclic whenever |Aut G| = φ(|G|). In an earlier paper we have answered both questions in the negative, and shown that |Aut G|/φ(|G|) can be made arbitrarily small. Here we show that these results remain true if G is restricted to being perfect, or soluble. The problem remains open when G is supersoluble, or nilpotent.
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