  In this paper, we consider the problem on the existence of perfect state
transfer(PST for short) on semi-Cayley graphs over abelian groups (which are
not necessarily regular), i.e on the graphs having semiregular and abelian
subgroups of automorphisms with two orbits of equal size. We stablish a
characterization of semi-Cayley graphs over abelian groups having PST. As a
result, we give a characterization of Cayley graphs over groups with an abelian
subgroup of index 2 having PST, which improves the earlier results on Cayley
graphs over abelian groups, dihedral groups and dicyclic group and determines
Cayley graphs over generalized dihedral groups and generalized dicyclic groups
having PST.
