I am wondering to what extent the functors "ring of invariants under a group action $G$" and "divided power envelope with respect to a $G$-stable ideal" commute.
To be precise, let $R$ be a commutative ring (with unit) and $G$ a group acting on $R$ by ring automorphisms. I am happy to assume that $R$ is a noetherian adic ring and that $G$ is profinite acting continuously on $R$, but I do not want to assume that $G$ is finite.
Let $S:=R^G$ be the subring of invariants, and suppose given an ideal $I\subseteq R$ which is $G$-stable. Denote by $J:=I\cap S$ the contraction of $J$ to $S$, and let $D(R,I)$ and $D(S,J)$ be the divided power envelopes of $R$ with respect to $I$ and $S$ with respect to $J$. (I should probably also complete these algebras with respect to a $G$-stable ideal of $R$ that is contained in the ideal of definition, so the induced $G$-action on the completed PD algebras is continuous).
Then $G$ acts on $D(R,I)$ and we have $D(S,J)\subseteq D(R,I)^G$.
Is this inclusion an equality?