Conjugacy classes of maximal cyclic subgroups

In this paper, we set $\eta (G)$ to be the number of conjugacy classes of maximal cyclic subgroups of $G$. We consider $\eta$ and direct and semi-direct products. We characterize the normal subgroups $N$ so that $\eta (G/N) = \eta (G)$\mathrm{.}Weset$G^- = { g \in G \mid \langle g \rangle {\rm ~is~not ~maximal~cyclic} }$\mathrm{.}Weshowif$\langle G^- \rangle < G$,then$G/\langle G^- \rangle$iseither(1)anelementaryabelian$p$-groupforsomeprime$p$, (2) a Frobenius group whose Frobenius kernel is a $p$-group of exponent $p$ and a Frobenius complement has order $q$ for distinct primes $p$ and $q$, or (3) isomorphic to $A_5$.