Groups satisfying the two-prime hypothesis with a composition factor isomorphic to ${\rm PSL}_2(q)$ for $q\geq 7$

Let $G$ be a finite group, and write ${\rm cd}(G)$ 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 $a, b \in {\rm cd}(G)$, the total number of (not necessarily different) primes of the greatest common divisor ${\rm gcd}(a, b)$ 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 ${\rm PSL}_2 (q)$ for $q \geq 7$.