An answer to a question of Kelarev and Praeger on Cayley graphs of semigroups

In Europ. J. Combinatorics 24 (2003) 59-72, Kelarev and Praeger posed a question: Is it true that if G is a semigroup with a subset S such that Cay (G, {s}) is Aut ({s}) (G)-vertex-transitive, for every s is an element of S, then the whole Cayley graph Cay (G, S) is ColAut(S) (G)-vertex-transitive, too? In this note, we give a negative answer to this problem and prove that in the cases of bands and completely simple semigroups the answer is positive.