Let $k$ be a field and let $\overline{k}$ be its algebraic closure. Let $K_0(Var/\overline{k})$ be the Grothendieck ring of algebraic varieties over $\overline{k}$.
Is it true that the natural action of $Gal(\overline{k}/k)$ on varieties over $\overline{k}$ descends to an action on $K_0(Var/\overline{k})$ by ring automorphisms? Has this action been studied somewhere, either in general or for some specific fields, for example, $k = \mathbb{Q}$ or $\mathbb{F}_p$?
