On the automorphism groups of smooth Fano threefolds

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.