  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.
