Minimal index of a $C^{*}$-crossed product by a finite group

Let $A\times_{\alpha}G$ be the $C^{*}$-crossed product of a simple unital $C^{*}$-algebra $A$ by a finite group $G$. In this paper we show that the canonical conditional expectation from $A\times_{\alpha}G$ to $A$ has the minimal index if $A\times_{\alpha}G$ is simple. It is also proved that if $\alpha$ is an outer action, then the canonical one is the unique conditional expectation of index-finite type from $A\times_\alpha G$ to $A$, while there are infinitely many conditional expectations when a nontrivial subgroup of $G$ acts innerly on $A$.