Extendability of automorphisms of K3 surfaces

A K3 surface X over a p-adic field K is said to have good reduction if it admits a proper smooth model over the ring of integers of K. Assuming this, we say that a subgroup G of Aut(X) is extendable if X admits a proper smooth model equipped with G-action (compatible with the action on X). We show that G is extendable if it is of finite order prime to p and acts symplectically (that is, preserves the global 2-form on X). The proof relies on birational geometry of models of K3 surfaces, and equivariant simultaneous resolutions of certain singularities. We also give some examples of non-extendable actions.