Centralizer of Sylow $p$-subgroups for nonabelian finite simple groups

Question

Given a nonabelian finite simple group $G$, is it always true that $$C_G(P)\leq P$$ for each Sylow $p$-subgroup $P\leq G$?

Answer 1

No, $P=\langle (12345)\rangle$ is a Sylow $5$-subgroup of $G=A_8$ but $(678)\in C_G(P)$.

Answer 2

For simple groups of lie type is true. This is Lemma 4 of M.R. Darafshe, Groups with the same non-commuting graph, Discrete Applied Mathematics, 157 (2009), pp. 833-837 doi: 10.1016/j.dam.2008.06.010.