On the second minimum average character degree of finite nonsolvable groups

Let $\mathrm{acd}\left(G\right)$ be the average character degree of a finite group G. It has been proved that $\min \left\{\mathrm{acd}\right(G\left)\right|G\in A\}=\mathrm{acd}({\text{A}}_5)=\frac{16}{5}$, where $A$ is the family of all finite nonsolvable groups. In this paper, we assume that $B$ is the family of all finite nonsolvable groups G having a nonabelian minimal normal subgroup not isomorphic to ${\text{A}}_5$. We prove that $\min \left\{\mathrm{acd}\right(G\left)\right|G\in B\}=\mathrm{acd}(\mathrm{PSL}(2,7))=\frac{14}{3}$. While we show that the second minimum average character degree of arbitrary nonsolvable groups does not exist, we classify all finite groups with $\mathrm{acd}\left(G\right)<14/3$.