Since ${\rm Sp}(2n,2)$ has trivial outer automorphism group and its natural module $M$ is absolutely irreducible, this follows from the fact that $|H^1({\rm Sp}(2n,2),M)| =2$.

You can find that result, for example, in Table 4.5 of 

Cohomology of finite groups of Lie type, I,
Edward Cline; Brian Parshall; Leonard Scott,
Publications Mathématiques de l'IHÉS (1975)
Volume: 45, page 169-191
(see [here](https://eudml.org/doc/103939)),

but that might not be the earliest proof.