Supersolvability and nilpotency in terms of the commuting probability and the average character degree

Let $p$ be a prime and let $G$ be a finite group such that the smallest prime that divides $\mathrm{\mid }G\mathrm{\mid }$ is $p$. We find sharp bounds, depending on $p$, for the commuting probability and the average character degree to guarantee that $G$ is nilpotent or supersolvable.