  In this article we offer the interpretation of the fermionic T-duality of the
type II superstring theory in double space. We generalize the idea of double
space doubling the fermionic sector of the superspace. In such doubled space
fermionic T-duality is represented as permutation of the fermionic coordinates
$\theta^\alpha$ and $\bar\theta^\alpha$ with the corresponding fermionic T-dual
ones, $\vartheta_\alpha$ and $\bar\vartheta_\alpha$, respectively. Demanding
that T-dual transformation law has the same form as inital one, we obtain the
known form of the fermionic T-dual NS-R i R-R background fields. Fermionic
T-dual NS-NS background fields are obtained under some assumptions. We conclude
that only symmetric part of R-R field strength and symmetric part of its
fermionic T-dual contribute to the fermionic T-duality transformation of
dilaton field and analyze the dilaton field in fermionic double space. As a
model we use the ghost free action of type II superstring in pure spinor
formulation in approximation of constant background fields up to the quadratic
terms.
