Let $S_\omega$ be the group of bijections $f:\omega\to\omega$, and let $F_\omega = \{\pi\in S_\omega: \exists N\in \omega(\pi(k) = k \text{ for all } k\geq N)\}$. It is easy to see that $F_\omega$ is a normal subgroup of $S_\omega$.
Is $S_\omega/F_\omega$ isomorphic to a subgroup of $S_\omega$?
No, it is not.
McKenzie (1971) observed that the "direct sum" of $\ge\aleph_1$ non-abelian groups cannot be embedded into $S_\omega$ (indeed, it yields an ascending chain of centralizers in $S_\omega$ of type $\omega_1$, and this is not possible).
On the other hand, the direct sum of $\aleph_1$ (or $2^{\aleph_0}$) countable groups can always be embedded in $S_\omega/F_\omega$: this easily follows from the fact that in a countable set there exist $2^{\aleph_0}$ infinite subsets with pairwise finite intersection.
