Action by automorphism group of a ring $R$ on $\mathrm{Spec}(R)$

Question

Let $R$ be a ring, and let $G\subset \mathrm{Aut}(R)$ be a subgroup of the automorphism group. Then $G$ also induces an action on the affine scheme $\mathrm{Spec}(R)$. Namely, for $g:R\to R$ an element in $G$, the action of $g$ on $\mathrm{Spec}(R)$ is just the isomorphism $$ \mathrm{Spec}(R)\to \mathrm{Spec}(R), \mathfrak p \to g^{-1}\mathfrak p $$

Let's simply consider the case $G=\mathrm{Aut}(R)$. Then,

Question: What should be the quotient $\mathrm{Spec}(R) /\mathrm{Aut}(R)$? Is there any interesting structure on this?

Answer 1

It's a general fact that for a group $G$ acting on an affine scheme $\operatorname{Spec} R$, we have that the quotient is given by the spectrum of the ring of invariants $R^G$. So the key is to figure out what $R^G$ is.

Determining what $R^{Aut(R)}$ seems to be a bit iffy, though - I had previously made an incorrect guess here, and don't see how to determine this in general for all suitable commutative rings.