  We prove that if $X$ is a smooth projective variety of dimension greater than
1 over a field $K$ of characteristic zero such that
$\operatorname{Pic}(X_{\bar{K}}) = \mathbb{Z}$ and $X_{\bar{K}}$ is simply
connected, then the natural map $\rho: \operatorname{Aut}(X) \to
\operatorname{Aut}(\operatorname{Sym}^d(X))$ is an isomorphism for every $d >
0$. We also partially compute the motivic zeta function of a Severi-Brauer
surface and explain some relations between the classes of Severi-Brauer
varieties in the Grothendieck ring of varieties.
