A note on smooth quotients of Prym varieties

We study pseudoreflections of geometric origin on Prym varieties of étale double covers. We prove that if the genus of the base curve is $g\ge 4$ then every such pseudoreflection has order 2. We use this result to show that, for $g\ge 5$, a non-trivial finite group $G$ of automorphisms of geometric origin acting faithfully on the Prym $P$ with $P\mathrm{/}G$ smooth must be isomorphic to either $\mathbb{Z}\mathrm{/}2\mathbb{Z}$ or $(\mathbb{Z}\mathrm{/}2\mathbb{Z}{)}^2$. We also show that the latter case can occur only for $g\le 7$. This sharpens results of Auffarth, Lahoz and Naranjo.