Equivariance and extendibility in finite reductive groups with connected center

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Characters of odd degree (2015, background): “Moreover, in the cases where the associated algebraic group has connected centre, Cabanes and the second author [CS13] could then verify the inductive McKay condition.”
The inductive Alperin–McKay and Blockwise Alperin Weight conditions for blocks with cyclic defect groups and odd primes (2013, unknown): “In a few cases the inductive conditions have been checked for specific finite simple groups and with respect to certain p-groups, see [36], [37], [38], and [11].”
Brauer's Height Zero Conjecture (2022, result): “The element s and its centralizer in G ∗ are described in terms of root systems in [3]. e e e Since e G ⋊ D induces Aut( G ) and [13, Thm 3.1] tells us Jordan decomposition for e G can be chosen to be D -equivariant, we then obtain as in the proof of Proposition 3.16 two non-Aut( G )-conjugate…”