In $G=\operatorname{PGL}(4,5)$ there are two elementary abelian $2$-subgroups of order $16$ denoted by $E_{1}$ and $E_{2}$ with $N_{G}(E_{1})=E_{1}.\operatorname{Sp}(4,2)$ and $N_{G}(E_{2})=E_{2}.(2^{3}:S_{3})$.

$N_{G}(E_{1})$ is a maximal subgroup of $G$ and $(2^{3}:S_{3})$ is a point stabilizer of a nonidentity element of $E_{2}$ in $\operatorname{Sp}(4,2)$ (viewing $E_{2}$ as a symplectic basis of $\operatorname{Sp}(4,2)$).

By comparison, $H=\operatorname{PGL}(4,3)$ has $N_{H}(F_{1})=F_{1}.M_{1}$ and $N_{H}(F_{2})=F_{2}.M_{2}$ where $F_{i}$ are again elementary abelian $2$-subgroups of order $16$ and $M_{i}$ are, by some Magma computation, two maximal subgroups of $\operatorname{Sp}(4,2)$ of orders $72$ and $120$, respectively. I see that $2^{4}.\operatorname{Sp}(4,2)$ is not a subgroup of $H$ any more, in contrast. (It appears in the corresponding finite unitary group now.) Also $M_{1}$ and $M_{2}$ are stabilizers in $\operatorname{Sp}(4,2)$ of $f_{1}\in F_{1}$ and $f_{2}\in F_{2}$ which have orbit sizes $10$ and $6$.

Is there an explanation for this phenomenon? What are these orbits of sizes $10$ and $6$ now? I hope to generalizer this. For instance, in $S=\operatorname{PGL}(8,3)$, what are the analogous orbit sizes in $2^6$?

I've only just found out that $M_{1}\cong \operatorname{SO}^{+}_{4}(2)$ and $M_{2}\cong \operatorname{SO}^{-}_{4}(2)$. They do have orbit sizes 1,5,10 and 1,9 6….

In $G=\operatorname{PGL}(4,5)$ there are two elementary abelian $2$-subgroups of order $16$ denoted by $E_{1}$ and $E_{2}$ with $N_{G}(E_{1})=E_{1}.\operatorname{Sp}(4,2)$ and $N_{G}(E_{2})=E_{2}.(2^{3}:S_{3})$.

$N_{G}(E_{1})$ is a maximal subgroup of $G$ and $(2^{3}:S_{3})$ is a point stabilizer of a nonidentity element of $E_{2}$ in $\operatorname{Sp}(4,2)$ (viewing $E_{2}$ as a symplectic basis of $\operatorname{Sp}(4,2)$).

By comparison, $H=\operatorname{PGL}(4,3)$ has $N_{H}(F_{1})=F_{1}.M_{1}$ and $N_{H}(F_{2})=F_{2}.M_{2}$ where $F_{i}$ are again elementary abelian $2$-subgroups of order $16$ and $M_{i}$ are, by some Magma computation, two maximal subgroups of $\operatorname{Sp}(4,2)$ of orders $72$ and $120$, respectively. I see that $2^{4}.\operatorname{Sp}(4,2)$ is not a subgroup of $H$ any more, in contrast. (It appears in the corresponding finite unitary group now.) Also $M_{1}$ and $M_{2}$ are stabilizers in $\operatorname{Sp}(4,2)$ of $f_{1}\in F_{1}$ and $f_{2}\in F_{2}$ which have orbit sizes $10$ and $6$.

Is there an explanation for this phenomenon? What are these orbits of sizes $10$ and $6$ now? I hope to generalizer this. For instance, in $S=\operatorname{PGL}(8,3)$, what are the analogous orbit sizes in $2^6$?

I've only just found out that $M_{1}\cong \operatorname{SO}^{+}_{4}(2)$ and $M_{2}\cong \operatorname{SO}^{-}_{4}(2)$. They do have orbit sizes 1,5,10 and 1,9,6….

In $G=\operatorname{PGL}(4,5)$ there are two elementary abelian $2$-subgroups of order $16$ denoted by $E_{1}$ and $E_{2}$ with $N_{G}(E_{1})=E_{1}.\operatorname{Sp}(4,2)$ and $N_{G}(E_{2})=E_{2}.(2^{3}:S_{3})$.

$N_{G}(E_{1})$ is a maximal subgroup of $G$ and $(2^{3}:S_{3})$ is a point stabilizer of a nonidentity element of $E_{2}$ in $\operatorname{Sp}(4,2)$ (viewing $E_{2}$ as a symplectic basis of $\operatorname{Sp}(4,2)$).

By comparison, $H=\operatorname{PGL}(4,3)$ has $N_{H}(F_{1})=F_{1}.M_{1}$ and $N_{H}(F_{2})=F_{2}.M_{2}$ where $F_{i}$ are again elementary abelian $2$-subgroups of order $16$ and $M_{i}$ are, by some Magma computation, two maximal subgroups of $\operatorname{Sp}(4,2)$ of orders $72$ and $120$, respectively. I see that $2^{4}.\operatorname{Sp}(4,2)$ is not a subgroup of $H$ any more, in contrast.(It appears in the corresponding finite unitary group now.) Also $M_{1}$ and $M_{2}$ are stabilizers in $\operatorname{Sp}(4,2)$ of $f_{1}\in F_{1}$ and $f_{2}\in F_{2}$ which have orbit sizes $10$ and $6$.

Is there an explanation for this phenomenon? What are these orbits of sizes $10$ and $6$ now? I hope to generalizer this. For instance, in $S=\operatorname{PGL}(8,3)$, what are the analogous orbit sizes in $2^6$?

I've only just found out that $M_{1}\cong \operatorname{SO}^{+}_{4}(2)$ and $M_{2}\cong \operatorname{SO}^{-}_{4}(2)$. They do have orbit sizes 1,5,10 and 1,9 6....

In $G=\operatorname{PGL}(4,5)$ there are two elementary abelian $2$-subgroups of order $16$ denoted by $E_{1}$ and $E_{2}$ with $N_{G}(E_{1})=E_{1}.\operatorname{Sp}(4,2)$ and $N_{G}(E_{2})=E_{2}.(2^{3}:S_{3})$.

$N_{G}(E_{1})$ is a maximal subgroup of $G$ and $(2^{3}:S_{3})$ is a point stabilizer of a nonidentity element of $E_{2}$ in $\operatorname{Sp}(4,2)$ (viewing $E_{2}$ as a symplectic basis of $\operatorname{Sp}(4,2)$).

By comparison, $H=\operatorname{PGL}(4,3)$ has $N_{H}(F_{1})=F_{1}.M_{1}$ and $N_{H}(F_{2})=F_{2}.M_{2}$ where $F_{i}$ are again elementary abelian $2$-subgroups of order $16$ and $M_{i}$ are, by some Magma computation, two maximal subgroups of $\operatorname{Sp}(4,2)$ of orders $72$ and $120$, respectively. I see that $2^{4}.\operatorname{Sp}(4,2)$ is not a subgroup of $H$ any more, in contrast.  (It appears in the corresponding finite unitary group now.) Also $M_{1}$ and $M_{2}$ are stabilizers in $\operatorname{Sp}(4,2)$ of $f_{1}\in F_{1}$ and $f_{2}\in F_{2}$ which have orbit sizes $10$ and $6$.

Is there an explanation for this phenomenon? What are these orbits of sizes $10$ and $6$ now? I hope to generalizer this. For instance, in $S=\operatorname{PGL}(8,3)$, what are the analogous orbit sizes in $2^6$?

I've only just found out that $M_{1}\cong \operatorname{SO}^{+}_{4}(2)$ and $M_{2}\cong \operatorname{SO}^{-}_{4}(2)$. They do have orbit sizes 1,5,10 and 1,9 6….