Professor Holt@ I think it is well known fact when an almost simple group $L$ with socle $K$ ($K$ is simple Lie type) contains an field or graph automorphism $x$ of prime order, then there exist an element $y\in xK$ with the above property. I hope this makes clear my motivation.

@Nick, I edit base on your last comment. Exept $PSL_2(q)$ is there any counterexample?

Thanks Geoff, Is there such a relation $|A|_p>|A|_r$ for almost simple group of lie type in characteristic p?

@Will, Thanks for your answer. Is there an explicit representation for these cases, specially case 2.