Finitary symmetric group is not fully invariant in symmetric group

Statement
The fact about::finitary symmetric group on an infinite set is not fully characteristic as a subgroup of the symmetric group.

Related facts

 * Finitary symmetric group is characteristic in symmetric group
 * Finitary symmetric group is strictly characteristic in symmetric group
 * Finitary symmetric group is not I-characteristic in symmetric group

Proof idea
We use the fact that an infinite set can be put in bijection with a union of countably many copies of that set. Using this bijection, we construct an endomorphism that sends each permutation to a permutation that acts like it on each of the countably many copies. This is an injective endomorphism, and under this endomorphism, every non-identity permutation gets mapped to a permutation that is not finitary. In particular, the finitary symmetric group is sent to a subgroup that it intersects trivially.