We study the fermionic T-duality symmetry of integrable Green-Schwarz sigma models on AdS backgrounds. We show that the sigma model on AdS₅ × S¹ background is self-dual under fermionic T-duality. We also construct new integrable sigma models on AdS₂ × CPⁿ. These backgrounds could be realized as supercosets of SU supergroups for arbitrary n, but could also be realized as supercosets of OSp supergroups for n = 1,3. We find that the supercosets based on SU supergroups are self-dual under fermionic T-duality, while the supercosets based on OSp supergroups are not. However, the reasons of OSp supercosets being not self-dual under fermionic T-duality are different. For OSp(6|2) case, corresponding to AdS₂ × CP³ background, the failure is due to the singular fermionic quadratic terms, just like AdS₄ × CP³ case. For OSp(3|2) case, the failure is due to the shortage of right number of κ-symmetry to gauge away the fermionic degrees of freedom, even though the fermionic quadratic term is not singular any more. More general, for the supercosets of the OSp supergroups with superalgebra B(n,m), including AdS₂ × S²ⁿ and AdS₄ × S²ⁿ backgrounds, the sigma models are not self-dual under fermionic T-duality as well, obstructed by the κ-symmetry.