Groups satisfying the two-prime hypothesis with a composition factor isomorphic to ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(q)$ for $q\ge 7$

Let $G$ be a finite group, and write $\mathrm{c}\mathrm{d}(G)$ for the degree set of the complex irreducible characters of $G$. The group $G$ is said to satisfy the {\it two-prime hypothesis} if, for any distinct degrees $a,b\in \mathrm{c}\mathrm{d}(G)$, the total number of (not necessarily different) primes of the greatest common divisor $\mathrm{g}\mathrm{c}\mathrm{d}(a,b)$ is at most $2$. In this paper, we prove an upper bound on the number of irreducible character degrees of a nonsolvable group that has a composition factor isomorphic to ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(q)$ for $q\ge 7$.