OK, your question makes sense now. The group  $2^7:{\rm Sp}(6,2)$ is a maximal subgroup of ${\rm Sp}(8,2)$, and is the stabilizer of a vector in the action of ${\rm Sp}(8,2)$ on its natural $8$-dimensional module. This immediately defines its action on that module. This action is the same in the split and nonsplit extensions
$2^8:{\rm  Sp}(8,2)$ and $2^8\cdot 
{\rm Sp}(8,2)$.