  Consider a smooth connected algebraic group $G$ acting on a normal projective
variety $X$ with an open dense orbit. We show that Aut($X$) is a linear
algebraic group if so is $G$; for an arbitrary $G$, the group of components of
Aut($X$) is arithmetic. Along the way, we obtain a restrictive condition for
$G$ to be the full automorphism group of some normal projective variety.
