We show that the decomposition matrix of a given group
G
G
is unitriangular, whenever
G
G
has a normal subgroup
N
N
such that the decomposition matrix of
N
N
is unitriangular,
G
/
N
G/N
is abelian and certain characters of
N
N
extend to their stabilizer in
G
G
. Using the recent result by Brunat–Dudas–Taylor establishing that unipotent blocks have a unitriangular decomposition matrix, this allows us to prove that blocks of groups of quasi-simple groups of Lie type have a unitriangular decomposition matrix whenever they are related via Bonnafé–Dat–Rouquier’s equivalence to a unipotent block. This is then applied to study the action of automorphisms on Brauer characters of finite quasi-simple groups. We use it to verify the so-called inductive Brauer–Glauberman condition that aims to establish a Glauberman correspondence for Brauer characters, given a coprime action.