Since ${\rm Sp}(2n,2)$ has trivial outer automorphism group, this follows from the fact that $|H^1({\rm Sp}(2n,2),M)| =2$, where $M$ is the natural module for ${\rm Sp}(2n,2)$ .
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),
but that might not be the earliest proof.