Let $p$ be a prime number. Denote by $\mathbb{Z}_p$ the ring of $p$-adic integers and let $R = \mathbb{Z}_p [[X_1, \ldots, X_n]] / (f_1, \ldots, f_d)$.
Assume that a finite abelian group $G$ of order prime to $p$ act non trivially on $R$. Let $I$ be the ideal of $R$ generated by the elements $g.r - r$ for all $g \in G$ and $r \in R$.
Consider the quotient map $f : R \to R/I$.
Is something known about the splitting of this map ? Namely, does there exist a $\mathbb{Z}_p$-algebra morphism $s : R/I \to R$ such that $f \circ s = id_{R/I}$?