Consider the simple group $A_\lambda$, the alternating group on the set $\lambda$, which I will assume has regular cardinality. Recall that this is the smallest subgroup of all permutations of $\lambda$ containing the finite even permutations. In particular, only finitely many elements of $\lambda$ are moved by elements of $A_\lambda$.
I would like to find a subgroup $L \lt A_\lambda$ such that $A_\lambda/L$ is countable. I considered using the subgroup that fixes a specific countable subset $\omega \hookrightarrow \lambda$, but from my back-of-envelope scribblings this didn't seem to work out - or I just couldn't see it!
Since $|A_\lambda| = \lambda$, we need $|L| = \lambda$ also, which puts constraints on what $L$ can be.