Let $G$ be a finite group. When does there exist a finite group $H$ such that every $h\in H$ is in the kernel of some epimorphism $H\to G$?


This is well-known to be true for $G$ abelian, for example $H=G\times G$ works. I would like very much to know such an $H$ for non-abelian $G$, e.g. for symmetric groups $G$.


Note that this is related to question "[Normal Covering of a Finite Group](https://mathoverflow.net/questions/185508/normal-covering-of-a-finite-group)"; there, it is discussed when $H$ is a union of proper normal subgroups (but not necessarily all with quotient $G$).



Covering finite groups by kernels
If $G$ is nonabelian simple, then $H$ cannot exist.


Lemma : Let $\cal X$ be a collection of normal subgroups of a group $H$ such that $\bigcap \cal X = 1$, and assume that $H/X$ is nonabelian simple for all $X \in \cal X$. Then $H$ is isomorphic to a direct product of nonabelian simple groups.


Proof: Let $M \triangleleft H$ be minimal such that $H/M$ is isomorphic to a direct product of nonabelian simple groups. To complete the proof, we show $M = 1$. Otherwise, there exists $X \in \cal X$ such that $M \cap X < M$, and we write $D = M \cap X$. Now
$M \not\subseteq X$, so $MX > X$, and since $H/X$ is simple, it follows that $MX = G$. Then $H/D \cong M/D \times X/D$, and this is a direct product of nonabelian simple groups since $M/D \cong H/X$ is nonabelian simple and $X/D \cong H/M$ is a direct product of nonabelian simples. This contradicts the minimality of $M$.QED


Theorem: A finite group $H$ cannot be covered by a collection $\cal X$ of normal subgroups such that $H/X$ is nonabelian simple for all $X \in \cal X$.


Proof: Assume $H$ actually can be so covered, and let $D = \bigcap \cal X$. Then $D \triangleleft H$ and $H/D$ is covered by the normal subgroups $X/D$, and each of these has a nonabelian simple quotient. We can thus replace $H$ by $H/D$, so we can assume $D = 1$. The lemma then applies, and $H$ is a direct product of a collection $\cal Y$ of nonabelian simple subgroups. Then every normal subgroup of $H$ is a product of some subset of $\cal Y$, and proper normal subgroups are products of proper subsets of $\cal Y$. Now choose one nonidentity element of each member of $\cal Y$ and let $h$ be the product of the chosen elements. Then $h$ lies in no proper normal subgroup of $H$, and this is a contradiction. 


