  We construct Cardy states in the Kazama-Suzuki model G/H x U(1), which
satisfy the boundary condition twisted by the automorphisms of the coset
theory. We classify all the automorphisms of G/H x U(1) induced from those of
the G theory. The automorphism group contains at least a Z_2 as a subgroup
corresponding to the charge conjugation. We show that in several models there
exist extra elements other than the charge conjugation and that the
automorphism group can be larger than Z_2. We give the explicit form of the
twisted Cardy states which are associated with the non-trivial automorphisms.
It is shown that the resulting states preserve the N=2 superconformal algebra.
As an illustration of our construction, we give a detailed study for two
hermitian symmetric space models SU(4)/SU(2) x SU(2) x U(1) and SO(8)/SO(6) x
U(1) both at level one. We also study the action of the level-rank duality on
the Cardy states and find the relation with the exceptional Cardy states
originated from a conformal embedding.
