# Is every group isomorphic to some nontrivial quotient group?

For any group $G$, does there exist a group $H$ and a nontrivial normal subgroup $N$ of $H$ such that $H/N\cong G$?

Yes, if $N$ (for nontrivial) is any nontrivial group (for instance the one with two elements), then the projection $H=G\times N\to G$ on the first factor has kernel $N\subseteq H$, so $G\cong H/N$.
Yes, for example $H:=G\times G$ and $N:=G\times e$
(if $G$ is trivial just take any nontrivial group $H$ and $H=N$, for example $H=N=\Bbb Z$)