Realization of positive rationals as |Aut(G)|/|G| for finite groups

Determine whether, for every positive rational number r, there exists a finite group G such that |Aut(G)|/|G| = r.

Background

The problem concerns characterizing which rational numbers can be realized as the ratio |Aut(G)|/|G|. A prior work posed this as an open problem, and the present paper resolves the abelian case negatively by showing strong restrictions (e.g., denominators must be squarefree and no odd prime can occur as the ratio), while also exhibiting that every power of 2 can occur.

The authors explicitly state that, notwithstanding their resolution for finite abelian groups, the general version for all finite groups remains open.