Let $X$ be a Brauer–Severi variety over $k$.

I understood that the automorphism scheme $\mathrm{Aut}_{X/k}=G$ as a group scheme acting on $X$ via $$(m,pr_2):G\times X \rightarrow X\times X$$ (the multiplication morphism), and that this morphism is surjective.

**Questions:**

What about the action of $\mathrm{Aut}_{X/k}(k)$ on $X$?

Is there for example a transitive action of $\mathrm{Aut}_{X/k}(k)$ on the closed points of $X$? If not, what kind of assumption is needed to get this?