The second minimum/maximum value of the number of cyclic subgroups of finite $p$-groups

Let $C(G)$ be the poset of cyclic subgroups of a finite group $G$ and let $\mathcal{P}$ be the class of $p$-groups of order $p^n$ ($n\ge 3$). Consider the function $\alpha :\mathcal{P}⟶(0,1]$ given by $\alpha (G)=\frac{\mathrm{\mid }C(G)\mathrm{\mid }}{\mathrm{\mid }G\mathrm{\mid }}$. In this paper, we determine the second minimum value of $\alpha$, as well as the corresponding minimum points. Further, since the problem of finding the second maximum value of $\alpha$ was completely solved for $p=2$, we focus on the case of odd primes and we outline a result in this regard.