A condition for distinguishing sceneries on non-abelian groups

A scenery f on a finite group G is a function from G to {0,1}. A random walk v(t) on G is said to be able to distinguish two sceneries if the distributions of the sceneries evaluated on the random walk with uniform initial distribution are identical only if one scenery is a shift of the other scenery. This paper generalizes a sufficient condition of Finucane, Tamuz, and Yaari for distinguishing two sceneries on finite abelian groups to one for finite non-abelian groups but shows that no random walks on finite non-abelian groups satisfy this sufficient condition.