Fix a prime number $p\neq 5$. Let $A$ be a finitely generated commutative associative unital $\mathbb{Z}[1/5]$-algebra. Assume the symmetric group $G=S_3$ acts on $A$ by algebra automorphisms. For a positive integer $n$ define the subalgebra $A_n\subset A$ $$ f\in A_n \iff \forall g\in G \quad g(f)-f\in p^n A. $$ Note that if $n\leq n'$ then $A_{n'}\subset A_{n}$.
Is it true that there exists a positive integer $n_0$ and a choice of generating elements $f_1, \dots, f_k$ of $A_{n_0}$ such that $A_n$ for any $n\geq n_0$ is generated by $p^{e_1} f_1, \dots, p^{e_k}f_k$ for some integers satisfying $0\leq e_i \leq n-n_0$?
In particular, if we denote by $m(n)$ the minimal number of generators of $A_n$, does there exist an integer $n_0$ such that $m(n)\leq m(n_0)$ for all $n\geq n_0$?
This question is not entirely arbitrary, I genuinely need it in a geometric computation. By Noether's results we know that $A_n$ are all finitely generated but I am not able to extract effective bounds.
For some specific $A$ in which I am interested, I can (via methods that do not generalize to other algebras) prove that this is true for $p>5$; $n_0$ can be taken to be $1$.