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

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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 $G\cong H/N$.

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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 $)

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