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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 ${\left(ℂX^{\mathrm{op}}\right)}^{*}⋊ℂG$. 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....