  The so-called inductive McKay condition on finite simple groups, due to
Isaacs-Malle-Navarro (2007), has been recently reformulated by Sp\"ath. We show
that this reformulation applies to the reduction theorem for Alperin's weight
conjecture, due to Navarro-Tiep (2011). This also simplifies the checking of
the inductive condition for Alperin's weight conjecture in the case of simple
groups of Lie type with regard to the defining prime.
