The Structure of $G\mathrm{/}\mathrm{\Phi }(G)$ in Terms of $\sigma (G)$

Let $G$ be a finite group. We let $\text{f}m(G)$ and $\text{sig}(G)$ denote the number of maximal subgroups of $G$ and the least positive integer $n$ such that $G$ is written as the union of $n$ proper subgroups, respectively. In this paper we determine the structure of $G\mathrm{/}\mathrm{\Phi }(G)$ when $G$ is a finite soluble group with $\text{f}m(G)\le 2\text{sig}(G)$.