Let $\mathbb{H}$ be the upper half of the complex plane, i.e., $\mathbb{H} =\{ z \in \mathbb{C}: \operatorname{Im} z >0\}$. And let $V$ be a Riemann surface with genus $g \ge 2$. Then $V$ is isomorphic to the quotient space $\mathbb{H}/\Gamma$ for some Fuchsian group $\Gamma$ with signature $(g;-)$. The total automorphism group $\operatorname{Aut} (V)$ of $V$ is isomorphic to $N(\Gamma)/\Gamma$, where $N(\Gamma)$ is the normalization of $\Gamma$ in $\operatorname{PSL}(2;\mathbb{R})$.
For any subgroup $G$ of $\operatorname{Aut} (V)$, there exists a corresponding subgroup $\Delta$ of $N(\Gamma)$ such that $G \cong \Delta /\Gamma$. Let $(h; m_1, \dots, m_b)$ be the signature of $\Delta$. Then $\Delta$ is generated by $2h$ hyperbolic elements and $b$ elliptic elements. Let $\gamma_1, \dots, \gamma_b$ be the elliptic generators of $\Delta$ with order $m_1, \dots, m_b$, respectively, let $P_i \in \mathbb{H}$ be the fixed point of $\gamma_i$, and let $Q_i \in W$ be the image of $P_i$ under the map $\mathbb{H} \rightarrow V \rightarrow W$. Now my questions are:
Is the genus of the quotient curve $W =V/G$ equal to $h$? Can we just write $W =\mathbb{H}/\Delta$?
Does the quotient map $\pi: V \rightarrow V/G =W$ have exactly $b$ branch points: $Q_1, \dots, Q_b$?
If the answer of question 2 is yes, around every ramification point $R_{i,j}$ of the map $\pi$ over the point $Q_i$, is $m_i$ equal to the ramification index of $\pi$ at $R_{i,j}$ (i.e., the map $\pi$ can be expressed as $z \mapsto z^{m_i}$)?
If the answers of the questions above are no, then what do the numbers $b$ and $m_i$'s mean in the map $\pi: V \rightarrow V/G =W$? Or if the answers are generally no, can they become true for some special case (say when the group $G$ satisfies some additional conditions like that $G$ is commutative, $G$ is cyclic group, and so on)?
I am reading Topics on Riemann Surfaces and Fuchsian Groups edited by Bujalance etc. Although I have not finish it, but it seems this book does not discuss the automorphism of Riemann surface too much. And I know another reference Discontinuous groups and birational transformations by Macbeath, but I really cannot find this book in the library or web. So I am also wondering if there is some reference easier to get on this topic.
I read one book long ago. I am not an expert in this area, but I also had came up at similar questions, and found following book interesting.
In the book Complex Functions: Algebraic and Geometric Veiw-point, the last one or two chapters discuss in detail (as per my understanding) the topics you mentioned. I hope that you will find it suitable and easier to study. But, it contains interesting exercises as well, which are important for understanding your questions.
There is book on Fuchsian groups, by Katok; but, I am not sure, whether it deals with surfaces (topologically and geometrically) or not.