  In this paper we are interested in the asymptotic enumeration of Cayley
graphs. It has previously been shown that almost every Cayley digraph has the
smallest possible automorphism group: that is, it is a digraphical regular
representation (DRR). In this paper, we approach the corresponding question for
undirected Cayley graphs. The situation is complicated by the fact that there
are two infinite families of groups that do not admit any graphical regular
representation (GRR).
  The strategy for digraphs involved analysing separately the cases where the
regular group $R$ has a nontrivial proper normal subgroup $N$ with the property
that the automorphism group of the digraph fixes each $N$-coset setwise, and
the cases where it does not. In this paper, we deal with undirected graphs in
the case where the regular group has such a nontrivial proper normal subgroup.
