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.