We start by giving an algorithm to construct conjugators in several groups of piecewise linear homeomorphisms. This algorithm, coupled with additional tweaks, returns a solution to the conjugacy problem in Thompson's group $F$. We will then recall a theorem of Bogopolski, Martino and Ventura which connects the conjugacy problem in group extensions to the twisted conjugacy problem in the base group. The algorithm we first give can be adapted to solve the twisted conjugacy problem, which requires knowledge of the automorphisms of $F$ (as described by Brin), and effectively allows us to create extensions of $F$ where the conjugacy problem is unsolvable. We will give all relevant definitions and constructions.