Degrees of $p$-rational characters and normality of Sylow $p$-subgroups

Several refinements of (the normality part of) the celebrated Itô--Michler theorem were obtained during the last two decades, in which the condition of having $p'$-degree, for a fixed prime $p$, is imposed only on some subsets of complex irreducible characters of a finite group $G$.

We prove further extensions of these results, where this condition is now imposed on the irreducible characters which lie above the principal character of a Sylow $p$-subgroup and are either $p$-rational, or strongly real when $p=2$.