Birational automorphism groups of Severi-Brauer surfaces over the field of rational numbers

We prove that the only non-trivial finite subgroups of birational automorphism group of non-trivial Severi–Brauer surfaces over the field of rational numbers are~$\mathbb{Z}\mathrm{/}3\mathbb{Z}$ and $(\mathbb{Z}\mathrm{/}3\mathbb{Z}{)}^2\mathrm{.}$ Moreover, we show that $(\mathbb{Z}\mathrm{/}3\mathbb{Z}{)}^2$ is contained in $\mathrm{B}\mathrm{i}\mathrm{r}(V)$ for any Severi–Brauer surface $V$ over a field of characteristic different from $2$ and $3$, and $(\mathbb{Z}\mathrm{/}3\mathbb{Z}{)}^3$ is contained in $\mathrm{B}\mathrm{i}\mathrm{r}(V)$ for any Severi–Brauer surface~$V$ over a field of characteristic different from $2$ and $3$ which contains a non-trivial cube root of unity.