The strong $\pi$-Sylow theorem for the groups PSL$_2(q)$

Let $\pi$ be a set of primes. A finite group $G$ is a $\pi$-group if all prime divisors of the order of $G$ belong to $\pi$. Following Wielandt, the $\pi$-Sylow theorem holds for $G$ if all maximal $\pi$-subgroups of $G$ are conjugate; if the $\pi$-Sylow theorem holds for every subgroup of $G$ then the strong $\pi$-Sylow theorem holds for $G$. The strong $\pi$-Sylow theorem is known to hold for $G$ if and only if it holds for every nonabelian composition factor of $G$. In 1979, Wielandt asked which finite simple nonabelian groups obey the strong $\pi$-Sylow theorem. By now the answer is known for sporadic and alternating groups. We give some arithmetic criterion for the validity of the strong $\pi$-Sylow theorem for the groups $\operatorname{PSL}_2(q)$.