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We show that the permutation module over $\mathbb{C}$ afforded by the actionof $Sp_{2m}(2^f)$ on its natural module is isomorphic to the permutation moduleover $\mathbb{C}$ afforded by the action of $Sp_{2m}(2^f)$ on the union of theright cosets of $O_{2m}^+(2^f)$ and $O_{2m}^-(2^f)$.

A remark on the permutation representations afforded by the embeddings of $O_{2m}^\pm(2^f)$ in $Sp_{2m}(2^f)$