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 $G$?