Galois subspaces for compact Riemann surfaces of genus 2

Let $X$ be a compact Riemann surface of genus 2 and $D$ a very ample divisor with ${\varphi }_D$ its associated embedding into ${\mathbb{P}}^n$. We consider the set $G_{X,D}$ of linear subspaces $W$ of ${\mathbb{P}}^n$ of codimension $2$ with projection ${\pi }_W$ such that $f_W={\pi }_W\circ {\varphi }_D$ is Galois, i.e. $f_W^{\ast }k({\mathbb{P}}^1)\subseteq k(X)$ is a Galois extension. It is known that $G_{X,D}$ is isomorphic to a disjoint union of projective spaces. In this article, we calculate the dimension of projective spaces in the decomposition of $G_{X,D}$, when $D$ is induced by a subgroup of $\mathrm{A}\mathrm{u}\mathrm{t}(X)$.