Ramification of covering maps of curves and the étale fundamental group I'm interested in finite galois covers $\varphi: Y \rightarrow X$ between smooth proper curves over an algebraically closed field of characteristic zero, which are étale outside a prescribed finite set $D$ of points of $X$.
More precisely, I'd like to understand how the étale fundamental group $\widehat\pi_1(U_{X})$ of $U_{X}:=X \setminus D$ determines the ramification indices $e_{y}$ of points $y$ of $Y$.
As a starting point I have the following more specific question:
Suppose that $X=\mathbb{P}^{1}_{k}$, where $k$ is an algebraically closed field of characteristic zero, and that $D=\{0,1,\infty\}$. The étale fundamental group of $U_{X}=\mathbb{P}^{1}_{k}\setminus \{0,1,\infty\}$ is then the profinite group generated by three elements $\sigma_1,\sigma_2,\sigma_3$, subject to the only relation
$$
\sigma_1 \cdot \sigma_2 \cdot \sigma_3=1
$$ Finite quotients of this group classify finite galois coverings $\varphi: Y \rightarrow \mathbb{P}^{1}_{k}$, which are étale outside of $\{0,1,\infty\}$.
Is it possible to choose the covering $\varphi:Y \rightarrow \mathbb{P}^{1}_{k}$ such that every point $y \in \varphi^{-1}(\{0,1,\infty\})$ has ramification index $e$, where $e$ is a fixed positive integer, $e \geq 2$? If so, which are the properties of the finite quotient of $\widehat{\pi}_1(\mathbb{P}^{1}_{k}\setminus\{0,1,\infty\})$, associated to $\varphi$, which reflect the order of the ramification indices?
I think that at least my first question might be related to (or possibly even be answered by) the Riemann existence theorem, but I was unable to figure out how.
 A: The relevant reference seems to be Exposé XIII, Corollaire 2.12. of SGAI. For the specific example $\Bbb{P}^1_k \setminus \{0,1,\infty\}$, with $k$ algebraically closed of characteristic zero, it says that giving a finite Galois cover
$$
\varphi: Y \rightarrow \Bbb{P}^1_k
$$
with (finite) Galois group $G$, which is étale outside of $\{0,1,\infty\}$, is equivalent to giving a surjective group homomorphism
$$
\phi: \widehat{\pi}_1(\Bbb{P}^1_k \setminus \{0,1,\infty\}) \rightarrow G
$$
Also it is shown in loc.cit. that $\widehat{\pi}_1(\Bbb{P}^1_k \setminus \{0,1,\infty\})$ is the pro-finite group generated by elements $\sigma_1,\sigma_2,\sigma_3$, subject only to the single relation
$$
\sigma_1 \cdot \sigma_2 \cdot \sigma_3=1
$$
Moreover, if we set $x_1=0$, $x_2=1$, $x_3=\infty$ (the actual ordering of the three points $0,1,\infty$ is irrelevant) then the ramification index of every point in $\varphi^{-1}(x_i)$ equals the order of $\phi(\sigma_i)$ in $G$.
So putting everything in context of the question, if I want a finite Galois cover 
$$\varphi: Y \rightarrow \Bbb{P}^1_k$$ which is étale outside of $D=\{0,1,\infty\}$ and has ramification index equal to $e$ at every point of $D$, I just have to find a finite group $G$, which can be generated by three elements $\overline{\sigma}_1,\overline{\sigma}_2,\overline{\sigma}_3$, each of order $e$ such that
$$
\overline{\sigma}_1 \cdot \overline{\sigma}_2 \cdot \overline{\sigma}_3 = 1
$$
It is clear that for a prime number $e \geq 3$, $G=\Bbb{Z}/e\Bbb{Z}$ does the job. This is in fact sufficient for my purposes.
