  In this paper we continue the study of group representations which are
counterexamples to the Ize conjecture. As in the previous papers by Lauterbach
[14] and Lauterbach & Matthews [15] we find new infinite series of finite
groups leading to such counterexamples. These new series are quite different
from the previous ones, for example the group orders do not form an arithmetic
progression. However, as before we find Lie groups which contain all these
groups. This additional structure was observed, but not used in the previous
studies of this problem.
  Here we also investigate the related bifurcations. To a large extent, these
are closely related to the presence of mentioned compact Lie group containing
the finite groups. This might give a tool to study the bifurcations related to
all low dimensional counterexamples of the Ize conjecture. It also gives an
indication of where we can expect to find examples where the bifurcation
behavior is different from what we have seen in the known examples.
