Joking aside...
Let $x: X \to \mathbb{P}^1$ be of degree $2$ and $\sigma: X \to X$ be an automorphism. Consider $f: X \to \mathbb{P}^1\times\mathbb{P}^1, f=(x,x\circ\sigma)$. If $f$ is injective, then $X$ embeds as a curve of bidegree $(2,2)$ in $\mathbb{P}^1\times\mathbb{P}^1$ and therefore has genus at most $1$. So, if $X$ has genus $>1$ then $f$ is $2-1$ and $f(X)$ has bidegree $(1,1)$ in $\mathbb{P}^1\times\mathbb{P}^1$, so $f(X)$ is the graph of an automorphism $\tau$ of $\mathbb{P}^1$ and $x\circ\sigma = \tau \circ x$. Finally $\tau$ has to send branch points to branch points because $\sigma$ sends ramification points to ramification points, as these points are Weierstrass points, and automorphisms preserve Weierstrass points.