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Let $G$ be a finite non-Dedekind $p$-group and $\nu^*(G)$ denote the number of conjugacy classes of non-normal cyclic subgroups of $G$. Does there exist a normal second maximal subgroup $S$ of $G$ such that $\nu^*(S)< \nu^*(G)$?

(A subgroup $S$ is a second maximal subgroup of the group $G$ if $S$ is maximal in a maximal subgroup $M$ of $G$. Since $G$ is a $p$-group, $S$ is a second maximal subgroup of $G$ if and only if $|G:S|=p^2$).

Any comment or answer will be appreciated!

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