I have a non-split extension $2^8\mathbin.(2^7\mathbin:\operatorname{Sp}(6,2))$ of $2^8$ by $2^7\mathbin:\operatorname{Sp}(6,2)$. The question is how does $2^7\mathbin:\operatorname{Sp}(6,2)$ act on $2^8$. This group sits maximally inside the unique nonsplit extension $2^8\mathbin.\operatorname{Sp}(8,2)$.



Group action w.r.t. non-split extension group of the form $2^8\mathbin.(2^7\mathbin:\operatorname{Sp}(6,2))$
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 $2^8\cdot 
{\rm Sp}(8,2)$.


