The question is related to this MO question:

Lifting varieties to characteristic zero.

Let $X$ be a projective smooth variety over $k$ alg. closed field of char. $p.$ Does there always exist an alteration $Y\to X,$ with $Y$ also projective smooth, such that $Y$ lifts to char. 0?

Side remark: over $k=\overline{\mathbb F}_p,$ modulo Tate conjecture, abelian varieties "generate" the motives of all proj. smooth varieties. Since abelian varieties are liftable, one can say that the (irred. components of) motives of any proj. smooth varieties is liftable in some sense. And I wonder if this can be realized geometrically. "Alteration" in my question is just a try; replace it with any reasonable geometric construction if you want. For instance, "a proper surjection" would be fine.