The Cayley theorem states that every group is isomorphic to a permutation group. This can be rephrased as follows: for each finite group $G$ there is a positive integer $n$ and a monomorphism from $G$ to $S_n$. Therefore the family $(S_n)_{n=1}^{\infty}$ is universal in the sense that any group is isomorphic to a subgroup of some family member.
Can we find a family with similar property, but with "epi" instead of "mono"? Ie., is there a reasonable family of finite groups, such that every finite group is a quotient group of some family member?
(By "reasonable" I mean something "smaller" than the family of all finite groups and not constructed specifically for our purpose.)