Finitely supported permutation ($\mathrm{supp}(\sigma )=\{n:\sigma (n)\ne n\}\text{ finite}$)

A finitely supported permutation fixes every element outside some finite subset. The finitely supported permutations of $\mathbb{N}$ are the union of the finite symmetric groups acting on $\{1,\dots ,k\}$, and hence form a countable set. All permutations of $\mathbb{N}$ form an uncountable set: independently swapping or fixing each pair $(2j-1,2j)$ injects the power set of $\mathbb{N}$ into this larger set.