Let X be a Brauer-Severi variety over k. I have understand that the automorphism scheme Aut_{X/k}=G as a group scheme acts on X via (m,pr_2):GxX /rightarrow XxX (multiplication morphism) and that this morphism is surjectiv.

But what is about the action of Aut_{X/k}(k) on X?

Is there for example a transitiv action of Aut_{X/k}(k) on the closed points of X?

Or what kind of assumption is needed to get this?