Irreducible characters of even degree and normal Sylow $2$-subgroups

The classical It\^o-Michler theorem on character degrees of finite groups asserts that if the degree of every complex irreducible character of a finite group $G$ is coprime to a given prime $p$, then $G$ has a normal Sylow $p$-subgroup. We propose a new direction to generalize this theorem by introducing an invariant concerning character degrees. We show that if the average degree of linear and even-degree irreducible characters of $G$ is less than $4/3$ then $G$ has a normal Sylow $2$-subgroup, as well as corresponding analogues for real-valued characters and strongly real characters. These results improve on several earlier results concerning the It\^o-Michler theorem.