For genus-6 curves, bielliptic curves descend to their field of moduli, and non-descent can only occur for curves on smooth degree-5 del Pezzo surfaces with automorphism group C2 or D10 (with √-1 outside the base field), or on 2A1-singular del Pezzo surfaces.
Field of moduli versus field of definition for cyclic covers of the projective line
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On the Fields of Moduli of Curves of Genus Six
For genus-6 curves, bielliptic curves descend to their field of moduli, and non-descent can only occur for curves on smooth degree-5 del Pezzo surfaces with automorphism group C2 or D10 (with √-1 outside the base field), or on 2A1-singular del Pezzo surfaces.