Have the groups "PSL(n,q)" and "PSL(n,q).f ", the same maxiaml abelian subgroups or not?(where "PSL(n,q).f " is the extension of PSL(n,q) by the field automorphism of it) Is there any counterexample for my question for example for some n,q?
$n=2, q=4$ is a counterexample: $PSL(2,4)=A_5$, the alternating group of degree $5$, which doesn't have abelian subgroups of order $\ge6$, and $PSL(2,4)\rtimes Aut(GF(4))=S_5$, which has an abelian subgroup of order $6$. These are probably the only counterexamples. Suggestion: Try to consider preimages of abelian groups in $SL(n,q)$, while they need not be abelian, maybe one can still play with Schur's Lemma and such. Also note that $SL(n,p^e)\rtimes Aut(GF(p^e))$ embeds into $GL_{ne}(p)$, which might be helpful. 

