Quotients of cubic surfaces

We consider the following problem: whether the quotient of cubic surface over algebraically non-closed field by a finite group of automorphisms is rational. We show, that the quotient can be non-rational only if the group is trivial or a cyclic group of order 3 acting on the cubic surface in definite way. Moreover, for this action we construct an explicit criterion of rationality of the quotient in terms of Galois group.