  Based on an effective 12-orbital tight-binding model, we examine the
superconducting states induced by the antiferromagnetic fluctuations for
iron-vacancy-ordered A$_{y}$Fe$_{2-x}$Se$_{2}$. It is shown that due to the
broken reflection symmetry induced by the iron vacancies, new superconducting
states with $C_{4h}$ symmetry emerge. In particular, we show that in the
$C_{4h}$ symmetry, symmetric axes of the pairing momenta do not need to
coincide with axes of the unit cell. As a result, in addition to the magnitude
of the pairing gap, the relative orientation of the pairing wave function to
the lattice forms another degree of freedom for characterizing the
superconducting gap and can further help in gaining the condensation energy.
Nonetheless, similar to other iron-based superconductors, the singlet ground
state is still dominated by s-wave or d-wave, which are nearly degenerate with
anisotropic gaps. Furthermore, s-wave and d-wave superconducting states are
separated by a quantum critical point controlled by the Hund's rule coupling
$J_{H}$.
