  As a main result of this paper we give conditions under which the generalized
$X$-join of Cayley graphs is a Cayley graph. In particular, we show that
$X$-join of isomorphic Cayley graphs is a Cayley grpah. To do this, new
properties for a generalized wreath product of permutation groups are given in
the case where the base group acts regularly. These are used to give conditions
for the generalized wreath product to contain a regular subgroup, which are
then applied to generalized $X$-joins of Cayley graphs. Along the way, it is
shown that the generalized $X$-join of isomorphic Cayley graphs will always be
a vertex transitive graph.
