Linear groups in Cremona groups

I shall discuss the following question. Let $G$ be a linear group, for example $\mathrm{GL}_n(k)$, where $k$ is a field or the ring of integers. When does $G$ embed into the Cremona group of birational transformations of a given variety $M$? Of course, this depends on $G$, on $M$, etc. Not much is known, but interesting problems show up.