  We study pseudo-simple heteroclinic cycles for a $\Gamma$-equivariant system
in $R^4$ with finite $\Gamma\subset O(4)$, 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
$\Gamma\subset O(4)$, $\Gamma\not\subset SO(4)$, admitting fragmentarily
asymptotically stable pseudo-simple heteroclinic cycles. (It has been
previously shown that for $\Gamma\subset SO(4)$ pseudo-simple cycles
generically are completely unstable.) Finally, we study a generalized
heteroclinic cycle, which involves a pseudo-simple cycle as a subset.
