  We give applications of integral canonical models of orthogonal Shimura
varieties and the Kuga-Satake morphism to the arithmetic of K3 surfaces over
finite fields. We prove every K3 surface of finite height over a finite field
admits a characteristic 0 lifting whose generic fiber is a K3 surface with
complex multiplication. Combined with the results of Mukai and Buskin, we prove
the Tate conjecture for the square of a K3 surface over a finite field. To
obtain these results, we construct an analogue of Kisin's algebraic group for a
K3 surface of finite height, and construct characteristic 0 liftings of the K3
surface preserving the action of tori in the algebraic group. We obtain these
results for K3 surfaces over finite fields of any characteristics, including
those of characteristic 2 or 3.
