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 of X fixes pointwise all the theta characteristics on X, then X is hyperelliptic with being its hyperelliptic involution.