Finitary symmetric group on subset is conjugacy-closed

Statement
Suppose $$S \subseteq T$$ are sets. Let $$\operatorname{FSym}(S)$$ and $$\operatorname{FSym}(T)$$ denote the finitary symmetric groups on $$S$$ and $$T$$ respectively, with $$\operatorname{FSym}(S)$$ viewed as a subgroup of $$\operatorname{FSym}(T)$$: any finitary permutation on $$S$$ corresponds to a finitary permutation on $$T$$ that acts the same way on $$S$$ and fixes $$T \setminus S$$ pointwise.

Then, $$\operatorname{FSym}(S)$$ is a conjugacy-closed subgroup of $$\operatorname{FSym}(T)$$: any two finitary permutations of $$S$$ that are conjugate in $$\operatorname{FSym}(T)$$ are also conjugate in $$\operatorname{FSym}(S)$$.