$C^{\ast }$-index of observable algebra in the field algebra determined by a normal group

Let $G$ be a finite group and $H$ a normal subgroup. $D(H;G)$ is the crossed product of $C(H)$ and $\mathbb{C}G$ which is only a subalgebra of $D(G)$, the quantum double of $G$. One can construct a $C^{\ast }$-subalgebra ${\mathcal{F}}_{{}_H}$ of the field algebra $\mathcal{F}$ of $G$-spin models, such that ${\mathcal{F}}_{{}_H}$ is a $D(H;G)$-module algebra. The concrete construction of $D(H;G)$-invariant subalgebra ${\mathcal{A}}_{{}_{(H,G)}}$ of ${\mathcal{F}}_{{}_H}$ is given. By constructing the quasi-basis of conditional expectation $z_{{}_H}$ of ${\mathcal{F}}_{{}_H}$ onto ${\mathcal{A}}_{{}_{(H,G)}}$, the $C^{\ast }$-index of $z_{{}_H}$ is given.