  We study genus $g$ hyperelliptic curves with reduced automorphism group $A_5$
and give equations $y^2=f(x)$ for such curves in both cases where $f(x)$ is a
decomposable polynomial in $x^2$ or $x^5$. For any fixed genus the locus of
such curves is a rational variety. We show that for every point in this locus
the field of moduli is a field of definition. Moreover, there exists a rational
model $y^2=F(x)$ or $y^2=x F(x)$ of the curve over its field of moduli where
$F(x)$ can be chosen to be decomposable in $x^2$ or $x^5$. While similar
equations have been given in Bujalance, Cirre, Gamboa and Gromadzki (2001) over
$\mathbb R$, this is the first time that these equations are given over the
field of moduli of the curve.
