  This is a contribution to the study of $\operatorname {Irr}(G)$ as an
$\operatorname {Aut}(G)$-set for $G$ a finite quasi-simple group. Focusing on
the last open case of groups of Lie type $\mathrm D$ and $^2\mathrm D$, a
crucial property is the so-called condition $A'(\infty)$ expressing that
diagonal automorphisms and graph-field automorphisms of $G$ have transversal
orbits in $\operatorname {Irr}(G)$. This is part of the stronger $A(\infty)$
condition introduced in the context of the reduction of the McKay conjecture to
a question on quasi-simple groups. Our main theorem is that a minimal
counter-example to condition $A(\infty)$ for groups of type $\mathrm D$ would
still satisfy $A'(\infty)$. This will be used in a second paper to fully
establish $A(\infty)$ for any type and rank. The present paper uses
Harish-Chandra induction as a parametrization tool. We give a new, more
effective proof of the theorem of Geck and Lusztig ensuring that cuspidal
characters of arbitrary standard Levi subgroups of $G={\mathrm D}_{
l,\mathrm{sc}}(q)$ extend to their stabilizers in the normalizer of that Levi
subgroup. This allows to control the action of automorphisms on these
extensions. From there Harish Chandra theory leads naturally to a detailed
study of associated relative Weyl groups and other extendibility problems in
that context.
