In the Kourovka notebook, Deaconescu asks whether $\gpord{\Aut G}\ge \phi(\gpord{G})$ for all finite groups $G$, where $\phi$ denotes the Euler totient function, and whether $G$ is cyclic whenever $\gpord{\Aut G}= \phi(\gpord{G})$. Both questions are answered in the negative in this paper. Moreover, $\gpord{\Aut G}/ \phi(\gpord{G)$ can be made arbitrarily small.