Schur rings over $\mathrm{S}\mathrm{p}(\mathbf{n},\mathbf{2})$ and multiplicity one subgroups

We study commutative Schur rings over the symplectic groups Sp$(n,2)$ containing the class $\mathcal{C}$ of symplectic transvections. We find the possible partitions of $\mathcal{C}$ determined by the Schur ring. We show how this restricts the possibilities for multiplicity one subgroups of Sp$(n,2)$.