We discuss the quantized, D = 4, non-Abelian BF theory within the supe
rfibre bundle formalism. By introducing an even pseudotensorial 2-supe
rform over a principal superfibre bundle with superconnection, we obta
in the geometrical BRST and anti-BRST transformations of the fields oc
curring in such a theory. These are the closed module classical equati
ons of motion. Then, giving a prescription to construct the gauge-fixi
ng action through a modified BRST operator, we determine the on-shell
nilpotent BRST symmetry and the quantum action. We also show how this
quantum action is invariant under the on-shell nilpotent anti-BRST sym
metry obtained by modifying the geometrical anti-BRST operator. Howeve
r, we find that, in relation to the anti-BRST symmetry, the quantum ac
tion also possesses a vector supersymmetry of ghost number (+1).