The Reconstruction of Cycle-free Partial Orders from their Abstract Automorphism Groups II : Cone Transitive CFPOs

In this triple of papers, we examine when two cycle-free partial orders can share an abstract automorphism group. This question was posed by M. Rubin in his memoir concerning the reconstruction of trees. In this middle paper, we adapt a method used by Shelah in \cite{ShelahPermutation} and \cite{ShelahPermutationErrata}, and by Shelah and Truss in \cite{ShelahTrussQuotients} to define a cone transitive CFPO inside its automorphism group using the language of group theory.