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 $I_\mathbb{N}$.

Turning arrows around, is there a group $S_\mathbb{N}$ with the following strong properties?

1. For every finite group $F$ there is a surjective group homomorphism $\pi:S_\mathbb{N}\to F$, and
2. If $G^*$ 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 $s:G^*\to S_\mathbb{N}$.

(Note that in the embedding case, there is no such group, see the comment section; in particular $I_\mathbb{N}$ is not "universal" in the above sense.)