Fields of moduli of hyperelliptic curves

Let F be an algebraically closed field with char(F) not equal to 2, let F/K be a Galois extension, and let X be a hyperelliptic curve defined over F. Let \iota be the hyperelliptic involution of X. We show that X can be defined over its field of moduli relative to the extension F/K if Aut(X)/<\iota> is not cyclic. We construct explicit examples of hyperelliptic curves not definable over their field of moduli when Aut(X)/<\iota> is cyclic.