On a character correspondence associated to $\mathfrak{F}$-projectors

We study the conditions under which the head characters of a finite solvable group, as defined by I. M. Isaacs, behave well with respect to restriction. We also determine the intersection of the kernels of all head characters of the group. Using G. Navarro’s definition of ${\mathfrak{F}}^{\mathrm{\prime }}$-characters, we generalize these results for any saturated formation $\mathfrak{F}$ containing the formation of nilpotent groups.