Automorphisms of curves fixing the order two points of the Jacobian

Let X be an irreducible smooth projective curve, of genus at least two, defined over an algebraically closed field of characteristic different from two. If X admits a nontrivial automorphism σ that fixes pointwise all the order two points of Pic0(X), then we prove that X is hyperelliptic with σ being the unique hyperelliptic involution. As a corollary, if a nontrivial automorphisms \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sigma^{\prime}}$$\end{document} of X fixes pointwise all the theta characteristics on X, then X is hyperelliptic with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sigma^{\prime}}$$\end{document} being its hyperelliptic involution.