Symmetric group on finite or cofinite subset is subset-conjugacy-closed

Statement
Suppose $$S \subseteq T$$ are sets. Then, the symmetric group $$\operatorname{Sym}(S)$$ embeds naturally as a subgroup of $$\operatorname{Sym}(T)$$: any permutation of $$S$$ extends to a permutation of $$T$$ as the identity map on $$T \setminus S$$.

With this embedding, if either $$S$$ or $$T \setminus S$$ is finite, $$\operatorname{Sym}(S)$$ is a conjugacy-closed subgroup in $$\operatorname{Sym}(T)$$. In other words, if two subsets $$A$$ and $$B$$ of $$\operatorname{Sym}(S)$$ are conjugate via an element $$g$$ of $$\operatorname{Sym}(T)$$, there is an element $$h$$ of $$\operatorname{Sym}(S)$$ such that $$hah^{-1} = gag^{-1}$$ for all $$a \in A$$.

Weaker facts

 * Symmetric group on finite or cofinite subset is conjugacy-closed