Partial orthogonal spreads over $\mathbb{F}_2$ invariant under the symmetric and alternating groups

Let m be an integer greater than 2 and let V be a vector space of dimension 2^m over F_2. Let Q be a non-degenerate quadratic form of maximal Witt index defined on V. We show that the symmetric group S_{2m+1} acts on V as a group of isometries of Q and permutes the members of a partial orthogonal spread of size 2m+1. This implies that any group of even order 2m or odd order 2m+1 acts transitively and regularly on a partial orthogonal spread in V. We also show that the alternating group A_9 acts in a natural manner on a complete spread of size 9 defined on a vector space of dimension 8 over F_2.