Extensions of characters in type D and the inductive McKay condition, II

We determine the action of the automorphism group Aut$(G)$ on the set of irreducible characters Irr$(G)$ for all finite quasi-simple groups $G$. For groups of Lie type, this includes the construction of an Aut$(G)$-equivariant Jordan decomposition of characters (Theorem B). We prove a property called $A(\mathrm{\infty })$ which includes an extendibility statement, known previously in types not $\mathrm{D}$ (Theorem A). Our methods blend here Shintani descent ideas introduced for type $\mathrm{B}$ along with an analysis of semisimple classes in the dual group $G^{\ast }$. The condition $A(\mathrm{\infty })$ originates in the program to prove the McKay conjecture using the classification of finite simple groups. Theorem C establishes the McKay conjecture for the prime 3.