For a finite group $G$ of Lie type and a prime $p$, the authors
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
$\mathrm{Out}(G)$ 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 $BG^\wedge _p$ in terms of
$\mathrm{Out}(G)$.
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