Rigid automorphisms of linking systems of finite groups of Lie type

Let $\mathcal{L}$ be a centric linking system associated to a saturated fusion system on a finite $p$-group $S$.

An automorphism of $\mathcal{L}$ is said to be rigid if it restricts to the identity on the fusion system.

An inner rigid automorphism is conjugation by some element of the center of $S$.

If $\mathcal{L}$ is the centric linking system of a finite group $G$, then rigid automorphisms of $\mathcal{L}$ are closely related to automorphisms of $G$ that centralize $S$.

For odd primes, all rigid automorphisms are known to be inner, but this fails for the prime 2.

We determine which known quasisimple linking systems at the prime 2 have a noninner rigid automorphism.

Based on previous results, this reduces to handling the case of the linking systems at the prime 2 of finite simple groups of Lie type in odd characteristic.

These have no noninner rigid automorphisms with two families of exceptions: the 2-dimensional projective special linear groups and even-dimensional orthogonal groups for quadratic forms of nonsquare discriminant.