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If $N$ and $H$ are finite groups, then there exist group $G$ s.t $N$ is normal in $G$ and $G/N\cong H$ (ex. $N\times H, N\rtimes H$ etc.)

Does there exist a group $G$ s.t. $N$ is characteristic subgroup of $G$ and $G/N\cong H$?

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    I deleted my answer because I misread the question.2011-08-27
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    @Geoff: It seemed to work to me, what was wrong?2011-08-27
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    The PO seems to want to fix both $H$ and $N.$ I thought $N$ was allowed to be chosen.2011-08-27
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    My gut feeling is that the answer is "no"; here's where I would look for a counterexample: there are $2$-groups $M$ and $K$ such that the "base" subgroup of the wreath product $M\wr K$ is not characteristic, and I would try $N=M$ and $H=K$. But I don't remember off-hand the structures of the groups $M$ and $K$, so I can't check.2011-08-28

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