I am trying to understand the following example, which I came across in a research article. I am posting it as a question below.

$\bf{Question}$. Let $\Sigma$ be a curve of genus two with the automorphism group $G$, and $p_1$, $p_2$, and $p_3$ are three points on $\Sigma$. Let $a_{1}$, $b_{1}$, $a_{2}$, $b_{2}$ denote the generators of $\pi_{1}(\Sigma)$ and $\gamma_{1}$, $\gamma_{2}$, $\gamma_{3}$ are simple loops about the points $p_{1}$, $p_{2}$, and $p_{3}$. Consider the following map $f$ from $\pi_{1} (\Sigma \setminus p_{1}, p_{2}, p_{3})$ to $\mathbb{Z}_{3}$ given by $\gamma_i \rightarrow 1$ and $a_{i}, b_{i} \rightarrow 0$. What it means the covering (unramified) $\Pi_{f} :\Sigma' \rightarrow \Sigma$ associated to the stabilizer of $f$? How one can think of such covering map in terms of the automorphism group $G$ of $\Sigma$? What one can say about the degree of $\Pi_{f}$?