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Suppose the finite group $N$ surjects to finite group $F$. It is true that for any $G = N ⋊_α \mathbb{Z}$ there are infinitely many covers of $G$ that are cyclic and surject to $F$.

But is this statement true for finitely generated $N$ or even general group $N$? Is there an article about it ?

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    $\begingroup$ Perhaps you could define exactly what you mean by a cyclic cover. And what is $Z$? $\endgroup$
    – Derek Holt
    Commented Mar 2, 2018 at 11:04
  • $\begingroup$ @DerekHolt Z is the set of integer. Sorry. $\endgroup$
    – Ma Joad
    Commented Mar 2, 2018 at 13:06
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    $\begingroup$ @MaJoad Derek asked you two questions, and you only answered one. Please answer his first question. $\endgroup$
    – YCor
    Commented Mar 2, 2018 at 22:53
  • $\begingroup$ @YCor OK. I will do it. $\endgroup$
    – Ma Joad
    Commented Mar 2, 2018 at 23:17

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