  Let $\mathcal{X}$ be a smooth Fano threefold over the complex numbers of
Picard rank $1$ with finite automorphism group. We give numerical restrictions
on the order of the automorphism group $\mathrm{Aut}(\mathcal{X})$ provided the
genus $g(\mathcal{X})\leq 10$ and $\mathcal{X}$ is not an ordinary smooth
Gushel-Mukai threefold. More precisely, we show that the order
$|\mathrm{Aut}(\mathcal{X})|$ divides a certain explicit number depending on
the genus of $\mathcal{X}$. We use a classification of Fano threefolds in terms
of complete intersections in homogeneous varieties and the previous paper of A.
Gorinov and the author regarding the topology of spaces of regular sections.
