Extra automorphisms of curves and definability over \bar Q
A generic elliptic curves over C has automorphism group of order 2. The elliptic curves with extra automorphisms are C/Z[i] (automorphism group of order 4) and C/Z[w] where w is a primitive 3rd root of unity (automorphism group of order 6). One can use their extra automorphisms to prove that they can be defined over Q
Similarly, the Klein quartic (a genus 3 curve with 168 automorphisms, the maximum possible number for genus 3 curves) can be defined over Q.
I feel like maybe one could try to approach this question by passing to the Jacobian of X and arguing that extra automorphisms of X give extra automorphisms of J(X) which means that J(X) has "large" endomorphism ring and hence must be defined over a number field (just as CM elliptic curves are defined over number fields), but I don't know whether "large" is large enough.