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Let $G=Sp(4,2^f)$ with $f>1$. Based on the facts when $f$ is small, I would feel the following:

$G$ has two conjugacy classes of subgroups isomorphic to $SO^+(4,2^f)$. One is in Aschbacher's class C8, and the other is $Sp(2,2^f)\wr2$ in C2. These two classes of subgroups are swapped by the graph automorphism of $G$.

Is this right and where can I find the related stuff? Thanks!

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  • $\begingroup$ It's important to specify here what you mean by "graph automorphism", which is special to characteristic 2. $\endgroup$
    – Jim Humphreys
    Commented Jul 30, 2013 at 17:39

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Yes it is right (for $f>1$), and it is (14.1)(2) of Aschbacher's paper

M. Aschbacher. On the maximal subgroups of the finite classical groups. Invent. Math. 76 (1984), 469–514.

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