It is true that $Sp_{2}(2)$ is a subgroup of the matrix group $\begin{pmatrix} Sp_2(2) & 0\\ *_{1\times 2} & 1\\ \end{pmatrix}$ where $*_{1\times2}$ denotes a $1\times 2$ matrix with arbitrary entries in the field of $2$ elements.
Question:
Is it true that $\begin{pmatrix} Sp_2(2) & 0\\ *_{2\times 2} & S_3\\ \end{pmatrix}$ is isomorphic to a subgroup of $\begin{pmatrix} Sp_2(2) & 0\\ *_{3\times 2} & S_4\\ \end{pmatrix}?$
It feels like true as $S_3 < S_4$ and $2^4 < 2^6$. I suppose the action on the $*$ parts wouldn't mess it up...
Side note: these groups are in fact the quotient groups $N_{G}(E)/C_{G}(E)$ where $G = PGL_{8}(13)$ and $E$ are elementary abelian $2$-subgroups of rank $2,3,4$ and $5$ respectively.
Analogous questions can be raised in $PGL_{n}(q)$. For example, in $PGL_{16}(5)$ there are quotient groups $N_{G}(E)/C_{G}(E)$ such as $\begin{pmatrix} Sp_2(2) & 0\\ *_{2\times 2} & S_3\\ \end{pmatrix}$ and $\begin{pmatrix} Sp_2(2) & 0\\ *_{7\times 2} & S_8\\ \end{pmatrix}$.