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Let k be a nonperfect field of characteristic $2$. Let $a\in k\backslash k^2$, $G=PGL_4(k)$ and and $$H= \left\{ \small\left[\begin{array}{cccc} x & ay &az & aw\\ w & x &ay & az \\ z & w & x & ay \\ y & z & w & x \\ \end{array} \right] \in PGL_4(k) \mid x, y, z, w \in k \right\}.$$ Can anybody find a connected proper overgroup or a connected proper subgroup of $H$ in $G$?

Let k be a nonperfect field of characteristic $2$. Let $a\in k\backslash k^2$, $G=PGL_4(k)$ and and $$H= \left\{ \small\left[\begin{array}{cccc} x & ay &az & aw\\ w & x &ay & az \\ z & w & x & ay \\ y & z & w & x \\ \end{array} \right] \in PGL_4(k) \mid x, y, z, w \in k \right\}.$$ Can anybody find a connected proper overgroup or a nontrivial connected proper subgroup of $H$ in $G$?

Let k be a nonperfect field of characteristic $2$. Let $a\in k\backslash k^2$, $G=PGL_4(k)$ and and $$H= \left\{ \small\left[\begin{array}{cccc} x & ay &az & aw\\ w & x &ay & az \\ z & w & x & ay \\ y & z & w & x \\ \end{array} \right] \in PGL_4(k) \mid x, y \in k \right\}.$$ Can anybody find a connected proper overgroup or a connected proper subgroup of $H$ in $G$?

Let k be a nonperfect field of characteristic $2$. Let $a\in k\backslash k^2$, $G=PGL_4(k)$ and and $$H= \left\{ \small\left[\begin{array}{cccc} x & ay &az & aw\\ w & x &ay & az \\ z & w & x & ay \\ y & z & w & x \\ \end{array} \right] \in PGL_4(k) \mid x, y, z, w \in k \right\}.$$ Can anybody find a connected proper overgroup or a connected proper subgroup of $H$ in $G$?

Let k be a nonperfect field of characteristic $2$. Let $a\in k\backslash k^2$. Let $G=PGL_4(k)$. Let $H=\{ \small\left[\begin{array}{cccc} x & ay &az & aw\\ w & x &ay & az \\ z & w & x & ay \\ y & z & w & x \\ \end{array} \right] \in PGL_4(k) \mid x, y \in k\}$. Can anybody find a connected proper overgroup or a connected proper subgroup of $H$ in $G$?

Let k be a nonperfect field of characteristic $2$. Let $a\in k\backslash k^2$, $G=PGL_4(k)$ and and $$H= \left\{ \small\left[\begin{array}{cccc} x & ay &az & aw\\ w & x &ay & az \\ z & w & x & ay \\ y & z & w & x \\ \end{array} \right] \in PGL_4(k) \mid x, y \in k \right\}.$$ Can anybody find a connected proper overgroup or a connected proper subgroup of $H$ in $G$?