Consider the divided-power ring $A := \mathbb Z \langle x_1, \ldots, x_n \rangle$ consisting of $\mathbb Z$-linear combinations of divided-power monomials of the form $x_1^{(a_1)} …

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, …