Finite quotients of powers of an elliptic curve

Let $E$ be an elliptic curve. When the symmetric group $\Sigma_{g+1}$ of order $(g+1)!$ acts on $E^{g+1}$ in the natural way, the subgroup $E_0^{g+1}$, consisting of those $(g+1)$-tuples whose coordinates sum to zero, is stable under the action of $\Sigma_{g+1}$. It is isomorphic to $E^g$. This paper concerns the structure of the quotient variety $E^g/\Sigma$ when $\Sigma$ is a subgroup of $\Sigma_{g+1}$ generated by simple transpositions. In an earlier paper we observed that $E^g/\Sigma$ is a bundle over a suitable power, $E^N$, with fibers that are products of projective spaces. This paper shows that $E^g/\Sigma$ has an \'etale cover by a product of copies of $E$ and projective spaces with an abelian Galois group.