  In this article, we will prove that the subsectors of $\alpha$-induced
sectors for $M \rtimes \hat{G} \supset M$ forms a modular category, where $M
\rtimes \hat{G}$ is the crossed product of $M$ by the group dual $\hat{G}$ of a
finite group $G$. In fact, we will prove that it is equivalent to M\"uger's
crossed product. By using this identification, we will exhibit an orbifold
aspect of the quantum double of $\Delta$(not necessarily non-degenerate)
obtained from a Longo-Rehren inclusion $A \supset B_\Delta$ under certain
assumptions.
  We will apply the above description of the quantum double of $\Delta$ to the
Reshetikhin-Turaev topological invariant of closed 3-manifolds, and we obtain a
simpler formula, which is a degenerate version of Turaev's theorem that the
Reshetikhin-Turaev invariant for the quantum double of a modular category
$\hat{\Delta}$ is the product of Reshetikhin-Turaev invariant of $\hat{\Delta}$
and its complex conjugate.
