  Part 1 : We remark that the conjugacy problem for pairs of hyperbolic au-
tomorphisms of a finitely presented group (typically a free group) is
decidable. The solution that we propose uses the isomorphism problem for the
suspensions, and the study of their automorphism group.
  Part 2 : In a previous work (part 1), we remarked that the conjugacy problem
for pairs of atoroidal automorphisms of a free group was solvable by mean of
the isomorphism problem for hyperbolic groups and an orbit problem for the
automorphism group of their suspensions (i.e. their semidirect product with Z
for the relevant structural automorphism). We consider the same problem a few
more automorphisms of free groups, those that produce relatively hyperbolic
suspensions that do not split over a parabolic subgroup.
