  We generalize Mukai and Shafarevich's definitions of isogenies between K3
surfaces over $\mathbb{C}$ to an arbitrary perfect field and describe how to
construct isogenous K3 surfaces over $\bar{\mathbb{F}}_p$ by prescribing linear
algebraic data when $p$ is large. The main step is to show that isogenies
between Kuga-Satake abelian varieties induce isogenies between K3 surfaces, in
the context of integral models of Shimura varieties. As a byproduct, we show
that every K3 surface of finite height admits a CM lifting under a mild
assumption on $p$.
