Factor group of the automorphism group of a matroid basis graph with respect to the automorphism group of the matroid

Any automorphism of a matroid induces an automorphism of its basis graph. We try to determine what can be said concerning the automorphisms of the basis graph which are not induced by matroids' automorphisms. In particular, we determine the structure of the factor group of the automorphism group of the basis graph with respect to the automorphism group of the matroid, in the event that this factor group exists.