A now-classical cyclic extension of the descent set of a permutation has been introduced by Klyachko and Cellini. Following a recent axiomatic approach to this notion, it is natural to ask which sets of permutations admit such an extension. The main result of this paper is a complete answer in the case of conjucay classes of permutations. It is shown that the conjugacy class of cycle type $λ$ has such an extension if and only if $λ$ is not of the form $(r^s)$ for some square-free $r$. The proof involves a detailed study of hook constituents in higher Lie characters.