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