  We construct an infinite family of triples (G, S1, S2) consisting of a group
G and a pair (S1,S2) of distinct subsets of G with the following properties. 1.
The distributions of the simple random walks on the two Cayley graphs Cay(G,
S1) and Cay(G, S2) are the same at an arbitrary specified moment of time, under
a permutation of the vertices. 2. The Spectra of Cay(G, Si) is decomposed into
a disjoint union of two subsets A and Bi of the equal size, which satisfies B1
= -B2.
