C*-basic construction between non-balanced quantum doubles

For finite groups X, G and the right G-action on X by group automorphisms, the non-balanced quantum double D(X; G) is defined as the crossed product (ℂX^op)* ⋊ ℂG. We firstly prove that D(X; G) is a finite-dimensional Hopf C*-algebra. For any subgroup H of G, D(X; H) can be defined as a Hopf C*-subalgebra of D(X; G) in the natural way. Then there is a conditonal expectation from D(X; G) onto D(X; H) and the index is [G; H]. Moreover, we prove that an associated natural inclusion of non-balanced quantum doubles is the crossed product by the group algebra.