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First of all, it's not true that any group can be realized as the commutator subgroup of some group. So, if we assume that there is atleast one group H with H' isomorphic to G, how to construct all such groups H?

To start with, we may assume that G is finite. Can we classify all (finite) groups with commutator subgroup isomorphic to G?

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    $\begingroup$ What is your motivation for the question.. just out of curiosity I am asking... $\endgroup$
    – Praphulla Koushik
    Commented Aug 23, 2018 at 16:24
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    $\begingroup$ I suppose you know that any finite Abelian group $G$ is isomorphic to $H^{\prime},$ where $H = G \wr \left( \mathbb{Z}/2\mathbb{Z} \right).$ Probably OK for infinite Abelian groups too. $\endgroup$
    – Geoff Robinson
    Commented Aug 23, 2018 at 16:26
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    $\begingroup$ @Geoff Thanks for this information. Didn't know this result! Can you please provide some reference where I can find the proof? $\endgroup$
    – Dey
    Commented Aug 23, 2018 at 16:32
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    $\begingroup$ @Dey you can rather prove it as an exercise: what is the derived subgroup of $G\wr C_2=(G\times G)\rtimes C_2$ (action by flip), for $G$ abelian? $\endgroup$
    – YCor
    Commented Aug 23, 2018 at 16:34
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    $\begingroup$ See the following post mathoverflow.net/questions/85540/… $\endgroup$
    – Alireza Abdollahi
    Commented Aug 29, 2018 at 18:58

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