Fix a finite field $k_p$ of characteristic $p$, as well as a $p$-adic lift of it, $k_0$. In other words $k_0$ is a $p$-adic field and $k_p$ its residue field.

Let $X_p$ be a non-singular variety over the finite field $k_p$.

The usual criterion for existence of lifts of $X_p$ to $k_0$ is the vanishing of obstructions in $H^2(X_p, \mathcal{T}_{X_p})$. When obstructions vanish, the space of lifts is a torsor under $H^1(X_p, \mathcal{T}_{X_p})$.

I am interested in learning about other kinds of existence criteria.

In particular if the $l$-adic cohomology of $X_p$ lifts as a Galois module to characteristic 0, what are additional obstructions to lifting of $X_p$ itself? How do dimensions of deformation spaces of the cohomological Galois representations compare with dimension of $H^1(X_p, \mathcal{T}_{X_p})$ etc.?

The case of Shimura varieties is of special interest.