  Let $C$ be a hyperelliptic curve of genus $g\geq 3$. In this paper we give a
new geometric description of the theta map for moduli spaces of rank 2
semistable vector bundles on $C$ with trivial determinant. In order to do this,
we describe a fibration of (a birational model of) the moduli space, whose
fibers are GIT quotients $(\mathbb{P}^1)^{2g}//PGL(2)$. Then, we identify the
restriction of the theta map to these GIT quotients with some explicit degree
two osculating projection. As a corollary of this construction, we obtain a
birational inclusion of a fibration in Kummer $(g-1)$-folds over $\mathbb{P}^g$
inside the ramification locus of the theta map.
