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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?

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  ?

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 cyclic covers of $G$ that surject to $F$.

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

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?

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 cyclic covers of $G$ that surject to $F$

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

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 cyclic covers of $G$ that surject to $F$.

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