  The purpose of this paper is to study hyperelliptic curves with extra
involutions. The locus $\L_g$ of such genus $g$ hyperelliptic curves is a
$g$-dimensional subvariety of the moduli space of hyperelliptic curves $\H_g$.
We discover a birational parametrization of $\L_g$ via dihedral invariants and
show how these invariants can be used to determine the field of moduli of
points $\p \in \L_g$.
  We conjecture that for $\p\in \H_g$ with $|\Aut(\p)| > 2$ the field of moduli
is a field of definition and prove this conjecture for any point $\p\in \L_g$
such that the Klein 4-group is embedded in the reduced automorphism group of
$\p$. Further, for $g=3$ we show that for every moduli point $\p \in \H_3$ such
that $| \Aut (\p) | > 4$, the field of moduli is a field of definition and
provide a rational model of the curve over its field of moduli.
