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