If I have understood the question correctly, then there seem to be lots of small counterexamples, such as $G=A_6, p=5$; $G=L_2(8)$ or $U_3(3)$, $p=7$; $G=M_{11}$ or $M_{12}$, $p=11$.

If I have understood the question correctly, then there seem to be lots of small counterexamples, such as $G=A_6, p=5$; $G=L_2(8)$ or $U_3(3)$, $p=7$; $G=M_{11}$ or $M_{12}$, $p=11$.

Added later: I thought of two more examples: $G=L_2(27), p=13$ and $G=L_3(5), p=31$. The interesting question is whether there are only finitely many examples. I would guess yes, but it could be hard to prove it.

If I have understood the question correctly, then there seem to be lots of small counterexamples, such as $G=A_6, p=5$; $G=L_2(8), p=7$; $G=M_{11}$ or $M_{12}$, $p=11$.

If I have understood the question correctly, then there seem to be lots of small counterexamples, such as $G=A_6, p=5$; $G=L_2(8)$ or $U_3(3)$, $p=7$; $G=M_{11}$ or $M_{12}$, $p=11$.