The continuity of $p$-rationality and a lower bound for $p^{\mathrm{\prime }}$-degree characters of finite groups

Let $p$ be a prime and $G$ a finite group. We propose a strong bound for the number of $p^{\mathrm{\prime }}$-degree irreducible characters of $G$ in terms of the commutator factor group of a Sylow $p$-subgroup of $G$. The bound arises from a recent conjecture of Navarro and Tiep [NT21] on fields of character values and a phenomenon called the continuity of $p$-rationality level of $p^{\mathrm{\prime }}$-degree characters. This continuity property in turn is predicted by the celebrated McKay-Navarro conjecture [Nav04]. We achieve both the bound and the continuity property for $p=2$.