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We prove that when  q  is a power of 2 every complex irreducible representation of   $\textrm{Sp}\big (2n, \mathbb{F}_{q}\big )$   may be defined over the real numbers, that is, all Frobenius–Schur indicators are 1. We also obtain a generating function for the sum of the degrees of the unipotent characters of   $\textrm{Sp}\big(2n, \mathbb{F}_{q}\big )$  , or of   $\textrm{SO}\big(2n+1,\mathbb{F}_{q}\big )$  , for any prime power  q .

Real Representations of Finite Symplectic Groups over Fields of Characteristic Two