The paper

D.G. Kharamtsov: Solvability of the conjugacy problem for finite subgroups in automorphism groups of free groups, Algebra and Logic 34(5), 1995, 311-337

solves the conjugacy problem for finite subgroups of ${\rm Aut}(G)$ and ${\rm Out}(G)$, but states that the conjugacy problem for elements of infinite order in these groups is still open, and that the "solution appears to be a goal not easy to be reached". -- I don't know whether this information is still up-to-date.

There does not seem to be an arXiv version of Martin Lustig's preprint Structure and conjugacy for automorphisms of free groups.

The paper

D.G. Kharamtsov: Solvability of the conjugacy problem for finite subgroups in automorphism groups of free groups, Algebra and Logic 34(5), 1995, 311-337

solves the conjugacy problem for finite subgroups of ${\rm Aut}(G)$ and ${\rm Out}(G)$, but states that the conjugacy problem for elements of infinite order in these groups is still open, and that the "solution appears to be a goal not easy to be reached". -- I don't know whether this information is still up-to-date.

There does not seem to be an arXiv- or even published version of Martin Lustig's preprint Structure and conjugacy for automorphisms of free groups.