The average character degree and an improvement of the Ito-Michler theorem

The classical It\^{o}-Michler theorem states that the degree of every ordinary irreducible character of a finite group $G$ is coprime to a prime $p$ if and only if the Sylow $p$-subgroups of $G$ are abelian and normal. In an earlier paper, we used the notion of average character degree to prove an improvement of this theorem for the prime $p=2$. In this follow-up paper, we obtain a full improvement for all primes.