  For a finite group G of Lie type and a prime p, we compare the automorphism
groups of the fusion and linking systems of G at p with the automorphism group
of G itself. When p is the defining characteristic of G, they are all
isomorphic, with a very short list of exceptions. When p is different from the
defining characteristic, the situation is much more complex, but can always be
reduced to a case where the natural map from Out(G) to outer automorphisms of
the fusion or linking system is split surjective. This work is motivated in
part by questions involving extending the local structure of a group by a group
of automorphisms, and in part by wanting to describe self homotopy equivalences
of the p-completion of the classifying space BG in terms of Out(G).
