Is it possible to find a genus two curve $C$ (over the field of complex numbers) with an endomorphism $\phi: C \to C$, such that $\phi$ has no fixed points and $\phi$ does not take any point to its hyperelliptic involution?



genus two curve with special automorphisms
No, $\phi$ must come from an automorphism of $\mathbb{P}^1$ and that has fixed points.


