We study pseudo-simple heteroclinic cycles for a Γ-equivariant system in R4 with finite Γ⊂O(4), and their nearby dynamics. In particular, in a first step towards a full classification – analogous to that which exists already for the class of simple cycles – we identify all finite subgroups of O(4) admitting pseudo-simple cycles. To this end we introduce a constructive method to build equivariant dynamical systems possessing a robust heteroclinic cycle. Extending a previous study we also investigate the existence of periodic orbits close to a pseudo-simple cycle, which depends on the symmetry groups of equilibria in the cycle. Moreover, we identify subgroups Γ⊂O(4), Γ⊄SO(4), admitting fragmentarily asymptotically stable pseudo-simple heteroclinic cycles. (It has been previously shown that for Γ⊂SO(4) pseudo-simple cycles generically are completely unstable.) Finally, we study a generalized heteroclinic cycle, which involves a pseudo-simple cycle as a subset.