Centralizers of automorphisms permuting free generators

By σ ∈ S _km we denote a permutation of the cycle-type k ^m and also the induced automorphism permuting subscripts of free generators in the free group Fk _m . It is known that the centralizer of the permutation σ in S _km is isomorphic to a wreath product Z _k / S _m and is generated by its two subgroups: the first one is isomorphic to Z _k ^m , the direct product of m cyclic groups of order k, and the second one is S _m . We show that the centralizer of the automorphism σ ∈ Aut(Fk _m ) is generated by its subgroups isomorphic to Z _k ^m and Aut(F _m ).