Isotropy of finite subset has finite double coset index in 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:


 * Isotropy of finite subset has finite double coset index in finitary symmetric group
 * Isotropy of finite subset has finite double coset index in finitary alternating group

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


 * Isotropy of a point has double coset index two in symmetric group
 * Isotropy of a point has double coset index two in finitary symmetric group
 * Isotropy of a point has double coset index two in finitary alternating group