  We study the Prime Graph Question for integral group rings. This question can
be reduced to almost simple groups by a result of Kimmerle and Konovalov. We
prove that the Prime Graph Question has an affirmative answer for all almost
simple groups having a socle isomorphic to $\operatorname{PSL}(2, p^f)$ for $f
\leq 2$, establishing the Prime Graph Question for the first time for all
automorphic extensions of series of simple groups. Using this, we determine
exactly how far the so-called HeLP-method can take us for (almost simple)
groups having an order divisible by at most $4$ different primes.
