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, $I_\mathbb{N}$ is countable. (I first thought that $I_\mathbb{N}$ embeds into every group having the above property -- I am wrong and I apologize. Side question: up to isomorphism, how many pairwise non-isomorphic countable groups $G$ are there such that every finite group embeds into $G$? - Not needed for acceptance of answer.)

Turning arrows around, is there a countable group $S_\mathbb{N}$ such that for every finite group $F$ there is a surjective group homomorphism $\pi:S_\mathbb{N}\to F$?

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.)

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.

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, $I_\mathbb{N}$ is countable. (I first thought that $I_\mathbb{N}$ embeds into every group having the above property -- I am wrong and I apologize. Side question: up to isomorphism, how many pairwise non-isomorphic countable groups $G$ are there such that every finite group embeds into $G$? - Not needed for acceptance of answer.)

Turning arrows around, is there a countable group $S_\mathbb{N}$ such that for every finite group $F$ there is a surjective group homomorphism $\pi:S_\mathbb{N}\to F$?

We say that a permutation $\varphi:\mathbb{N}\to\mathbb{N}$ is almost-identical 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 almost identical 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 $\pi:S\to F$, 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.

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.