  In this paper we will study perturbative quantum gravity on supermanifolds
with both noncommutative and non-anticommutative coordinates. We shall first
analyses the BRST and the anti-BRST symmetries of this theory. Then we will
also analyze the effect of shifting all the fields of this theory in background
field method. We will construct a Lagrangian density which apart from being
invariant under the extended BRST transformations is also invariant under
on-shell extended anti-BRST transformations. This will be done by using the
Batalin-Vilkovisky (BV) formalism. Finally, we will show that the sum of the
gauge-fixing term and the ghost term for this theory can be elegantly written
down in superspace with two Grassmann parameters.
