  We investigate a class of odd (ramification) coverings $C \to \mathbb{P}^1$
where $C$ is hyperelliptic, its Weierstrass points maps to one fixed point of
$\mathbb{P}^1$ and the covering map makes the hyperelliptic involution of $C$
commute with an involution of $\mathbb{P}^1$. We show that the total number of
hyperelliptic odd coverings of minimal degree $4g$ is ${3g \choose g-1} 2^{2g}$
when $C$ is general. Our study is approached from three main perspectives: if a
fixed effective theta characteristic is fixed they are described as a solution
of a certain class of differential equations; then they are studied from the
monodromy viewpoint and a deformation argument that leads to the final
computation.
