Do gonal morphisms have non-trivial automorphisms

Question

Let $X$ be a compact connected Riemann surface.

Let $\pi:X\to \mathbf{P}^1$ be a gonal morphism, i.e., a morphism of minimal degree.

Can $\pi$ have non-trivial automorphisms? (An automorphism of $\pi$ is an automorphism of $\sigma:X\to X$ of $X$ such that $\pi\circ \sigma = \pi$.)

Is this true if $\pi$ is hyperelliptic, i.e., if $\deg \pi = 2$ and $g\neq 1$?

Answer 1

If $\pi$ is a Galois cover (e.g. when $X$ is hyperelliptic), then the set of the automorphisms of $\pi$ is the Galois group of the cover, thus is non-trivial if $\deg \pi >1$.