Finitary alternating group is conjugacy-closed in symmetric group

Statement
Suppose $$S$$ is an infinite set, $$G$$ is the symmetric group on $$S$$, and $$H$$ is a subgroup of $$G$$ comprising the even finitary permutations, i.e., the finitary alternating group on $$S$$. Then, $$H$$ is a conjugacy-closed subgroup of $$G$$: if two elements of $$H$$ are conjugate in $$G$$, they are conjugate in $$H$$.

In particular, $$H$$ is a fact about::conjugacy-closed normal subgroup of $$G$$.

Related facts

 * Finitary symmetric group is conjugacy-closed in symmetric group
 * Finitary alternating group is monolith in symmetric group
 * Finitary alternating group is characteristic in symmetric group