  We study the fermionic T-duality symmetry of integrable Green-Schwarz sigma
models on AdS backgrounds. We show that the sigma model on $AdS_5\times S^1$
background is self-dual under fermionic T-duality. We also construct new
integrable sigma models on $AdS_2\times CP^n$. 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 $AdS_2\times CP^3$ background, the
failure is due to the singular fermionic quadratic terms, just like
$AdS_4\times CP^3$ case. For $OSp(3|2)$ case, the failure is due to the
shortage of right number of $\kappa$-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 $AdS_2\times S^{2n}$ and $AdS_4\times S^{2n}$
backgrounds, the sigma models are not self-dual under fermionic T-duality as
well, obstructed by the $\kappa$-symmetry.
