Let
G
G
be a finite solvable group. We construct a set
H
\mathcal {H}
of irreducible characters of
G
G
such that if
C
C
is a Carter subgroup of
G
G
, then the members of
H
\mathcal {H}
behave well with respect to
C
C
-composition series for
G
G
, and we show that
H
\mathcal {H}
is in bijective correspondence with the set of linear characters of
C
C
. Also, if
A
A
is a group that acts coprimely on
G
G
, then analogously, we characterize in terms of
A
A
-composition series for
G
G
, the set of
A
A
-invariant characters of
G
G
that have a linear Glauberman-Isaacs correspondent.