Automorphisms of symmetric powers and motivic zeta functions

We prove that if $X$ is a smooth projective variety of dimension greater than 1 over a field $K$ of characteristic zero such that $\mathrm{Pic}(X_{\bar{K}})=\mathbb{Z}$ and $X_{\bar{K}}$ is simply connected, then the natural map $\rho :\mathrm{Aut}(X)\to \mathrm{Aut}({\mathrm{Sym}}^d(X))$ 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.