It is well-known that if $A$ is an ordinary abelian variety over a finite perfect field $ k$ of characteristic $ p>0$ and $ W=W(k)$ is the ring of Witt vectors over $ k$, then the canonical lifting $ A_{can}$ of $A$ to $W$ is characterized by the fact that every endomorphism $f$ of $A$ lifts to and endomorphism of $A_{can}$. In other words, the natural map $ End(A_{can})----> End(A)$ is bijection.

Now is there a characterization of CM liftings of abelian varities (not necessarily ordinary) through liftings of endomorphisms? in particular is there a characterization of CM liftings based on liftings of Frobenius?