  In this note we look at the moduli space $\cR_{3,2}$ of double covers of
genus three curves, branched along 4 distinct points. This space was studied by
Bardelli, Ciliberto and Verra. It admits a dominating morphism $\cR_{3,2} \to
{\mathcal A}_4$ to Siegel space. We show that there is a birational model of
$\cR_{3,2}$ as a group quotient of a product of two Grassmannian varieties.
This gives a proof of the unirationality of $\cR_{3,2}$ and hence a new proof
for the unirationality of ${\mathcal A}_4$.
