Quotients of cubic surfaces

Let $\mathbb{k}$ be any field of characteristic zero, $X$ be a cubic surface in ${\mathbb{P}}_{\mathbb{k}}^3$ and $G$ be a group acting on $X$. We show that if $X(\mathbb{k})\mathrm{\ne }\mathrm{\varnothing }$ and $G$ is not trivial and not a group of order $3$ acting in a special way then the quotient surface $X\mathrm{/}G$ is rational over $\mathbb{k}$. For the group $G$ of order $3$ we construct examples of both rational and nonrational quotients of both rational and nonrational $G$-minimal cubic surfaces over $\mathbb{k}$.