We construct Cardy states in the Kazama-Suzuki model G/H×U(1), which satisfy the boundary condition twisted by the automorphisms of the coset theory. We classify all the automorphisms of G/H×U(1) induced from those of the G theory. The automorphism group contains at least a ℤ2 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 ℤ2. 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)×SU(2)×U(1) and SO(8)/SO(6)×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.