Thank you for your comment! Your way proposed here is actually to do discrete Fourier transform to the vector $(z_0,z_1,\dots,z_{m-1})$. I've tried this transform before but followed by trying some uncertainty principle. Now I'll try exactly your way and bless myself:-)

@DerekHolt:$N_G((PSL_2(16)\times PSL_2(16)).2^2)$ is maximal in $G$ if and only if $G=L$ or $L.2$, so by your computation for $\Gamma O_8^+(4)$, my statement holds. Thank you so much!

@DerekHolt: What if $G$ contains a graph automorphism?

@DerekHolt:By THEOREM A of Liebeck, Praeger and Saxl, The maximal factorizatins of the finite simple groups and their automorphism groups, I would be able to show that if $H$ is not contained in any maximal parabolic subgroup then $H\cap L$ is contained in $(L_2(16)\times L_2(16)).2^2$ (as $C_2$ or $C_3$ subgroup of $L$). Does this help?