Quotients of Severi–Brauer surfaces

We show that a quotient of a non-trivial Severi–Brauer surface $S$ over arbitrary field $\mathbb k$ of characteristic 0 by a finite group $G\subset\operatorname{Aut}(S)$ is $\mathbb k$-rational if and only if $|G|$ is divisible by 3. Otherwise, the quotient is birationally equivalent to $S$.