  Let $p$ be a prime and $G$ a finite group. We propose a strong bound for the
number of $p'$-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 $p'$-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 $p=2$.
