  We introduce a new approach of computing the automorphism group and the field
of moduli of points $\p=[C]$ in the moduli space of hyperelliptic curves
$\H_g$. Further, we show that for every moduli point $\p \in \H_g(L)$ such that
the reduced automorphism group of $\p$ has at least two involutions, there
exists a representative $C$ of the isomorphism class $\p$ which is defined over
$L$.
