  A rigid automorphism of a linking system is an automorphism which restricts
to the identity on the Sylow subgroup. A rigid inner automorphism is
conjugation by an element in the center of the Sylow subgroup. At odd primes,
it is known that each rigid automorphism of a centric linking system is inner.
We prove that the group of rigid outer automorphisms of a linking system at the
prime $2$ is elementary abelian, and that it splits over the subgroup of rigid
inner automorphisms. In a second result, we show that if an automorphism of a
finite group $G$ restricts to the identity on the centric linking system for
$G$, then it is of $p'$-order modulo the group of inner automorphisms, provided
$G$ has no nontrivial normal $p'$-subgroups. We present two applications of
this last result, one to tame fusion systems.
