Descent equalities and the inductive McKay condition for types B and E

We establish the inductive McKay condition introduced by Isaacs-Malle-Navarro \cite{IMN} for finite simple groups of Lie types ${\text{tB}}_l$ ($l\ge 2$), ${\text{tE}}_6$, ${}^2{\text{tE}}_6$ and ${\text{tE}}_7$, thus leaving open only the types $\text{tD}$ and ${}^2\text{tD}$. We bring to the methods previously used by the authors for type $\text{tC}$ \cite{CS17C} some descent arguments using Shintani’s norm map. This provides for types different from $\text{tA},\text{tD},{}^2\text{tD}$ a uniform proof of the so-called global requirement of the criterion given by the second author in \cite[2.12]{S12}. The local requirements from that criterion are verified through a detailed study of the normalizers of relevant Levi subgroups and their characters.