Realizations of groups of piecewise continuous transformations of the circle

We study the near action of the group \begin{document}$ \mathrm{PC} $\end{document} of piecewise continuous self-transformations of the circle. Elements of this group are only defined modulo indeterminacy on a finite subset, which raises the question of realizability: a subgroup of \begin{document}$ \mathrm{PC} $\end{document} is said to be realizable if it can be lifted to a group of permutations of the circle. We prove that every finitely generated abelian subgroup of \begin{document}$ \mathrm{PC} $\end{document} is realizable. We show that this is not true for arbitrary subgroups, by exhibiting a non-realizable finitely generated subgroup of the group of interval exchanges with flips. The group of (oriented) interval exchanges is obviously realizable (choosing the unique left-continuous representative). We show that it has only two realizations (up to conjugation by a finitely supported permutation): the left and right-continuous ones.