Covering monolithic groups with proper subgroups

Given a finite non-cyclic group $G$, call $\sigma (G)$ the smallest number of proper subgroups of $G$ needed to cover $G$. Lucchini and Detomi conjectured that if a nonabelian group $G$ is such that $\sigma (G)<\sigma (G\mathrm{/}N)$ for every non-trivial normal subgroup $N$ of $G$ then $G$ is $\mathit{\text{monolithic}}$, meaning that it admits a unique minimal normal subgroup. In this paper we show how this conjecture can be attacked by the direct study of monolithic groups.