  We study bipartite distance-regular Cayley graphs with diameter three or
four. We give sufficient conditions under which a bipartite Cayley graph can be
constructed on the semidirect product of a group -- the part of this bipartite
Cayley graph which contains the identity element -- and $\mathbb{Z}_{2}$. We
apply this to the case of bipartite distance-regular Cayley graphs with
diameter three, and consider cases where the sufficient conditions are not
satisfied for some specific groups such as the dihedral group. We also extend a
result by Miklavi\v{c} and Poto\v{c}nik that relates difference sets to
bipartite distance-regular Cayley graphs with diameter three to the case of
diameter four. This new case involves certain partial geometric difference sets
and -- in the antipodal case -- relative difference sets.
