On the conjugacy problem for cyclic extensions of free groups
  We study the conjugacy problem in cyclic extensions of free groups. It is shown that the conjugacy problem is solvable in split extensions of finitely generated free groups by virtually inner automorphisms. An algorithm for construction of the unique representative (the conjugacy normal form) for each conjugacy class is given. 
Valerij Bardakov, Leonid Bokut, Andrei Vesnin