For a finite non-cyclic group $G$ we say that a collection $\mathcal H$ of proper subgroups of $G$ is a *cover* if $G = \bigcup_{H \in \mathcal H} H$. If $H^g \in \mathcal H$ for any $H \in \mathcal H$ and $g \in G$, we say that $\mathcal H$ is a *normal cover*. One can now define $\gamma(G)$ to be the minimal number of conjugacy classes in a normal cover of $G$. This number has been investigated in a couple of papers.
One could restrict the subgroups in the collections $\mathcal H$ to have additional properties, for example one could require them to be abelian or cyclic. In the latter case we would then count the number of conjugacy classes of maximal cyclic subgroups of $G$ (let us denote this number of $\gamma_c(G)$).

Has the quantity $\gamma_c(G)$ been investigated thoroughly in the literature? I have only found very little information.