The McKay correspondence establishes a connection between the set of finite subgroups of $\mathrm{SU}(2)$ and the set of affine Dynkin diagrams.
Some of these diagrams are very symmetric; for instance, the group of graph automorphisms of the Dynkin diagram $D_4$ is isomorphic to the symmetric group $S_4$.
How does the existence of graph automorphisms on the Dynkin side of the correspondence translate (if it does) to a representation theoretic statement about the corresponding subgroup of $\mathrm{SU}(2)$?