Let $X$ be an absolutely irreducible reduced surface over a finite field $k$ of characteristic $p$. What are sufficient and necessary conditions for $X$ to be a generalized Zariski surface over $k$ (there is a purely inseparable rational dominant map $\varphi\!: \mathbb{P}^2 \dashrightarrow X$ over $k$)?

More precisely, I am interested in the case when $X$ is the Kummer surface of an Jacobian $J$, which is also a generalized Zariski surface over the algebraic closure $\overline{k}$ (there is a purely inseparable rational dominant map $\psi\!: \mathbb{P}^2 \dashrightarrow X$ only over $\overline{k}$).