Let $X$ be a smooth projective variety over a finite field, with a closed immersion to some other smooth projective variety $S$.

Suppose there exists an fpqc cover $S'\to S$, such that $X\times_SS'$ can be lifted to mixed characteristic.

> Can $X$ be lifted to mixed characteristic?

For example, suppose $S = \mathbb{P}^n$, and $X\to S$ is a projective embedding. Take $S^{\rm perf}$, the perfection of $S$ (inverse limit along the absolute Frobenius map). Then $S' := S^{\rm perf}\to S$ is an fpqc cover.

> Say $X_{S'}$ is liftable to mixed characteristic. Is $X$?
Equivalently, do obstructions to liftability of $X$ give obstructions to liftability of $X_{S'}$?

My guess is that the answer is no. If anyone sees an obstruction to liftability of $X_{S'}$ coming from non-liftability of $X$, that is what I am looking for.