  We show that the decomposition matrix of a given group $G$ is unitriangular,
whenever $G$ has a normal subgroup $N$ such that the decomposition matrix of
$N$ is unitriangular, $G/N$ is abelian and certain characters of $N$ extend to
their stabilizer in $G$. 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
Bonnaf\'e--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 {\it inductive
Brauer--Glaubermann condition}, that aims to establish a Glauberman
correspondence for Brauer characters, given a coprime action.
