For a finite symmetry group G of an aperiodic substitution tiling system (,ω), we show that the crossed product of the tiling C*-algebra by G has real rank zero, tracial rank one, a unique trace, and that order on its K-theory is determined by the trace. We also show that the action of G on satisfies the weak Rokhlin property, and that it also satisfies the tracial Rokhlin property provided that has tracial rank zero. In the course of proving the latter we show that is finitely generated. We

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... o provide a link between and the AF algebra Connes associated to the Penrose tilings.