Let $f:\mathbb{N}\rightarrow\mathbb{N}$ be nondecreasing such that $f(n+1)-f(n)$ is bounded. Let $S_\mathbb{N}$ be the bijections from $\mathbb{N}$ to $\mathbb{N},$ i.e. the symmetric group of $\mathbb{N}.$ Let $$G_f=\{\phi\in S_\mathbb{N}|\exists m\in\mathbb{N}:-mf(\phi_n)\leq\phi_n-n\leq mf(n)\text{ eventually always}\}.$$ In simpler terms, these are the permutations $\phi$ such that $\phi_n$ and $\phi^{-1}_n$ stray away from $n$ at a rate of $O(f(n))$ (it turns out that this is an exact characterization after some proof). I was able to prove that $G_f$ is always a permutation group in $S_\mathbb{N}$ and there are uncountably many distinct $G_f.$ Has anyone seen these groups in the literature of the infinite symmetric group? They seem to be very useful in computing rearrangements of conditionally convergent sums, but I haven't been able to find them anywhere.
Note: If we replace the $f(n+1)-f(n)$ is bounded assumption with $f(n)=O(n)$ and $n=O(f(n)),$ then we still have that $G_f$ is a group.