Automorphism groups of higher-dimensional sphere graphs

Determine whether the automorphism group of the d-dimensional sphere is isomorphic to the automorphism group of the d-sphere graph for every integer d>1.

Background

The paper proves in dimension one that the automorphism group of the unit circle is isomorphic to the automorphism group of its circle graph. For the higher-dimensional analogue, it asks whether the same correspondence holds between the automorphism group of the d-sphere and that of the associated d-sphere graph.