  We investigate how the fixed point algebra of a C*-dynamical system can
differ from the underlying C*-algebra. For any exact group $\Gamma$ and any
infinite group $\Lambda$, we construct an outer action of $\Lambda$ on the
Cuntz algebra $\mathcal{O}_2$ whose fixed point algebra is almost equal to the
reduced group C*-algebra ${\rm C}^\ast_{\rm r}(\Gamma)$. Moreover, we show that
every infinite group admits outer actions on all Kirchberg algebras whose fixed
point algebras fail the completely bounded approximation property.
