De Bruijn digraphs and shuffle-exchange graphs are useful models for interconnection networks. They can be represented as group action graphs of the wrapped butterfly graph and the cube-connected cycles, respectively. The Kautz digraph has the similar definitions and properties to de Bruijn digraphs. It is d-regular and strongly d-connected, thus it is a group action graph. In this paper, we use another representation of the Kautz digraph and settle the open problem posed by M.-C. Heydemann in [ G. Hahn (ed.) et al., Graph symmetry: algebraic methods and applications. Proceedings of the NATO Advanced Study Institute and séminaire de mathématiques supérieures, Montréal, Canada, July 1-12, 1996. Dordrecht: Kluwer Academic Publishers. NATO ASI Ser., Ser. C, Math. Phys. Sci. 497, 167–224 (1997; Zbl 0885.05075)].