Which are the with respect to inclusion largest subgroups $G < {\rm Sym}(\mathbb{N})$ such that every finitely generated subgroup $H$ of $G$ for which the natural density of the set of integers $n$ whose orbit under the action of $H$ contains no integer less than $n$ exists and is zero has only finitely many orbits on $\mathbb{N}$?

We say that a subgroup of ${\rm Sym}(\mathbb{N})$ has sparse orbit representatives if it has infinitely many orbits on $\mathbb{N}$, but the set of smallest orbit representatives has natural density 0 (and in particular its natural density exists).

Which are the with respect to inclusion largest subgroups of ${\rm Sym}(\mathbb{N})$ which do not have finitely generated subgroups with sparse orbit representatives?