  We solve the twisted conjugacy problem on Thompson's group F. We also exhibit
orbit undecidable subgroups of Aut(F), and give a proof that Aut(F) and
Aut_+(F) are orbit decidable provided a certain conjecture on Thompson's group
T is true. By using general criteria introduced by Bogopolski, Martino and
Ventura in [5], we construct a family of free extensions of F where the
conjugacy problem is unsolvable. As a byproduct of our techniques, we give a
new proof of a result of Bleak-Fel'shtyn-Goncalves in [4] showing that F has
property R_\infty, and which can be extended to show that Thompson's group T
also has property R_\infty.
