Fano threefolds of genus 12 with large automorphism group in positive and mixed characteristic

We study prime Fano threefolds of genus 12 ($V_{22}$-varieties) with positive-dimensional automorphism groups in positive and mixed characteristic. We classify such varieties over any perfect field. In particular, we prove that $V_{22}$-varieties of Mukai-Umemura type over $k$ exist if and only if $\mathrm{char}\ k \neq 2$, $5$. We also prove the same result for $\mathbb{G}_a$-type. As arithmetic applications, we show that the Shafarevich conjecture holds for $V_{22}$-varieties of Mukai-Umemura type and of $\mathbb{G}_m$-type, while it fails for $V_{22}$-varieties of $\mathbb{G}_a$-type. Moreover, we prove that there exists $V_{22}$-varieties over $\mathbb{Z}$, whereas there do not exist $V_{22}$-varieties over $\mathbb{Z}$ whose generic fiber has a positive-dimensional automorphism group.