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 $g\neq 1$?