The conjugacy problem in extensions of Thompson's group

In a recent paper, Bogopolski, Martino and Ventura develop a criterion to study the conjugacy problem in extension of groups. This is achieved by reducing to the study of two other decision problems: the twisted conjugacy problem and the orbit decidability problem. We describe a simplified point of view of the conjugacy in Thompson's group F which allows us to attack these decision problems. We produce examples of extensions of F where the conjugacy problem is solvable and others where it is unsolvable. We also prove related results on twisted conjugacy class and discuss possible generalizations. This is joint work with Jose' Burillo and Enric Ventura.