Extendability of group actions on K3 or Enriques surfaces

Abstract

Let $X$ be a K3 or Enriques surface with good reduction. Let $G$ be a finite group acting (not necessarily linearly) on $X$. We give a criterion for this group action to extend to a smooth model of $X$ in terms of the action of $G$ on the second $\ell$-adic cohomology groups. In particular, we generalize the result on the extendability of Galois actions on K3 surfaces by Chiarellotto, Lazda, and Liedtke. As an application, we prove that a symplectic linear group action is extendable if the residue characteristic does not divide its order. Lastly, we relate the good reduction of Enriques surfaces with that of their K3 double covers.