Galois unirational surfaces

A surface $X$ is called unirational if there exists a dominant rational map $\mathbb{P}^2\dashrightarrow X$. For algebraically closed fields of characteristic zero any unirational surface is rational by Castelnuovo’s rationality criterion. But if the ground field $\mathbb k$ is an arbitrary algebraically non-closed field then this does not hold. For example, any del Pezzo surface of degree $4$, $3$, or $2$ (having a point defined over $\mathbb k$ in sufficiently general position) is unirational over $\mathbb k$, but some of these surfaces are not rational over $\mathbb k$. A particular case of unirational surfaces is Galois unirational surfaces, that are surfaces birationally equivalent to quotients of rational surfaces. In the talk I will give a complete classification of Galois unirational surfaces over algebraically non-closed fields of characteristic zero, and discuss some related questions.