Let a finite group [Formula: see text] act coprimely on a finite group [Formula: see text]. The Glauberman–Isaacs correspondence [Formula: see text] is a bijection from the set of [Formula: see text]-invariant irreducible characters of [Formula: see text] onto the set [Formula: see text] of irreducible characters of the centralizer of [Formula: see text] in [Formula: see text]. Let [Formula: see text] be a subgroup of [Formula: see text]. Composing from left to right, it follows that [Formula: see text] is an injection from [Formula: see text] into [Formula: see text]. We show that, in some cases, the map can be defined via the actions of some subgroups of [Formula: see text] containing [Formula: see text] on the centralizers in [Formula: see text] of some other such subgroups. We also show in many instances, such as [Formula: see text] odd or [Formula: see text] supersolvable and [Formula: see text] solvable, that this map is independent of the overgroup [Formula: see text].