Let $D_{n}=\langle a,b \ | a^n=b^2=abab=e\rangle$ be a dihedral group. Assume that $b$ acts on $\mathbb{P}^1$ by $z\mapsto \overline{z}$ where $z$ is an inhomogeneous coordinate of $\mathbb{P}^1$. Assume also that $a$ acts on $\mathbb{P}^1$ "holomorphically". What action of $D_{2n}$ on $\mathbb{P}^1$ is possible?
I am aware of the action $a:z\rightarrow e^{\frac{2\pi i k}{n}}z$ for $1\le k \le n$. This yields non-faithful action of $D_{2n}$, as the action of $a$ and $b$ commute. Are there any other actions?