We say that a permutation $\varphi:\mathbb{N}\to\mathbb{N}$ is finitary if there is $k\in\mathbb{N}$ such that $\varphi(i) = i$ for all $i\in\mathbb{N}$ with $i\geq k$. Let $I_\mathbb{N}$ denote the group of finitary permutations of $\mathbb{N}$, with composition as group operation. Every finite group can be embedded into $I_\mathbb{N}$.

Moreover, $I_\mathbb{N}$ has the universal property that if $G$ is such that every finite group can be embedded into $G$, then $I_\mathbb{N}$ embeds into $G$.

Turning arrows around, is there a group $S_\mathbb{N}$ such that

1. for every finite group $F$ there is a surjective group homomorphism $\pi:S_\mathbb{N}\to F$, and
2. whenever $G$ is a group such that for every finite group $F$ there is a surjective group homomorphism $\pi:G\to F$, then there is a surjective group homomorphism $\pi':G\to S_\mathbb{N}$?

Also, it would be interesting to know if $S_\mathbb{N}$ is unique up to isomorphism.