  We consider the two generalizations of lamplighter groups: automata groups
generated by Cayley machine and cross-wired lamplighter groups. For a finite
step two nilpotent group with central squares, we study its associated Cayley
machine and give a presentation of the corresponding automata group. We show
the automata group is a cross-wired lamplighter group and does not embed in the
wreath product of a finite group with a torsion free group. For a subfamily of
such finite step two nilpotent groups, we prove that their associated automata
groups are linear.
