Counting Non-abelian Coverings of Algebraic Curve

In this article, we study the etale coverings of an algebraic curve $C$ with Galois group a semi-direct product $\mathbb{Z}\mathrm{/}m\mathbb{Z}⋊\mathbb{Z}\mathrm{/}n\mathbb{Z}$. Especially, for a given etale cyclic $n$-covering $D\to C$, we determine how many curves $E$ are there, satisfying $E\to D$ is an etale cyclic $m$-covering and $E\to C$ is Galois with non-abelian Galois group, under the assumption $gcd(m,n)=1$.