Good reduction of K3 surfaces

Let K be the field of fractions of a local Henselian discrete valuation ring OK of characteristic zero with perfect residue field k. Assuming potential semi-stable reduction, we show that an unramified Galois action on the second l-adic cohomology group of a K3 surface over K implies that the surface has good reduction after a finite and unramified extension. We give examples where this unramified extension is really needed. Moreover, we give applications to good reduction after tame extensions and Kuga-Satake Abelian varieties. On our way, we settle existence and termination of certain ops in mixed characteristic, and study group actions and their quotients on models of varieties.