An equivariant Whitehead algorithm and conjugacy for roots of Dehn twist automorphisms

Given finite sets of cyclic words {u(1), ..., u(k)} and {nu (1), ..., nu (k )} in a finitely generated free group F and two finite groups A and B of ou ter automorphisms of F, we produce an algorithm to decide whether there is an automorphism which conjugates A to B and takes u(i) to nu (i) for each i . If 4 and B are trivial, this is the classic algorithm due to Whitehead. W e use this algorithm together with Cohen and Lustig's solution to the conju gacy problem for Dehn twist automorphisms of F to serve the conjugacy probl em for outer automorphisms which have a power which is a Dehn twist. This s ettles the conjugacy problem for all automorphisms of F which have linear g rowth.