CM-lifts for Isogeny Classes of Shimura F-crystals over Finite Fields

We extend to large contexts pertaining to Shimura varieties of Hodge type a result of Zink on the existence of lifts to characteristic 0 of suitable representatives of certain isogeny classes of abelian varieties endowed with Frobenius and other endomorphisms over ${\text{dbF}}_{p^q}$, whose $p$-divisible groups in mixed characteristic $(0,p)$ are with complex multiplication.