Isotropy of finite subset has finite double coset index in symmetric group

This article gives the statement, and possibly proof, of a particular subgroup or type of subgroup satisfying a particular subgroup property (namely, Subgroup of finite double coset index (?)) in a particular group or type of group (namely, Symmetric group (?)).

Statement

There exists a function $f: \mathbb{N} \to \mathbb{N}$ with the following property. Let $S$ be a set and $A$ be a subset of $S$ of size $n$. Let $G$ be the symmetric group on $S$ and $H$ be the subgroup of $G$ comprising permutations that fix every point of $S$. Then, the double coset index of $H$ in $G$ is at most $f(n)$. Further, if the cardinality of $S$ is at least $2n$, the double coset index of $H$ in $G$ is precisely $f(n)$.

Further, we have: $(n + 1)! \le f(n) \le (2n)!$.

Related facts

Some analogous statements for the finitary symmetric group and the finitary alternating group:

In the special case where the subset has size one, we get a subgroup of double coset index two: