20.26 (2022)

Let $\phi$ be the Euler totient function. Does there exist a constant $a > 0$ such that $|\text{Aut}(G)| \geqslant \phi(|G|)^a$
$\qquad$ a) for every finite group $G$?
$\qquad$ b) for every finite nilpotent group $G$?

If such a constant $a$ exists, then $a \leqslant \frac{40}{41}$ (J. González-Sánchez, A. Jaikin-Zapirain, Forum Math. Sigma, 3, Article ID e7, 11 p., electronic only (2015)). Also cf. 12.77.