Non-liftability of automorphism groups of a K3 surface in positive characteristic

We show that a characteristic $0$ model $X_R\to \Spec R$, with Picard number $1$ over a geometric generic point, of a K3 surface in characteristic $p\ge 3$, essentially kills all automorphisms (Theorem 5.1). We show that there is an explicitely constructed automorphism on a supersingular K3 surface in characteristic $3$, which has positive entropy, the logarithm of a Salem number of degree $22$ (Theorem 6.4). In particular it does not lift to characteristic $0$. In addition, we show that in any large characteristic, there is an automorphism of a supersingular K3 which has positive entropy and does not lift to characteristic $0$ (Theorem 7.5).