Perfect state transfer (PST) has great significance due to its applications in quantum information processing and quantum computation. The main problem we study in this paper is to determine whether the two-fold Cayley tree, an extension of the Cayley tree, admits perfect state transfer between two roots using quantum walks. We show that PST can be achieved by means of the so-called nonrepeating quantum walk [Phys. Rev. A 89 042332 (2014)] within time steps that are the distance between the two roots; while both the continuous-time quantum walk and the typical discrete-time quantum walk with Grover coin approaches fail. Our results suggest that in some cases the dynamics of a discrete-time quantum walk may be much richer than that of the continuous-time quantum walk.