I want to know the outer automorphisms of the Weyl group of $\mathrm{E}_8$, if any.

But I might as well ask the question more generally. Suppose we have a Coxeter diagram. This gives a Coxeter group. What are the outer automorphisms of this group? It seems we get one from any symmetry of the diagram; are these all of them?

(If we were forming the simply connected compact Lie group $G$ from a Dynkin diagram, we'd know every outer automorphism of $G$ comes from a symmetry of the diagram. So, I'm hoping this analogous result is true. But maybe it's too bold a generalization; I'll settle for Coxeter diagrams that come from Dynkin diagrams. The Dynkin diagram of $\mathrm{E}_8$ has no symmetries, of course.)