On the Mext groups of sVec_R and sVec_H

We compute the groups of minimal nondegenerate extensions of the real symmetric fusion categories $\mathrm{sVec}_{\mathbb R}$ and $\mathrm{sVec}_{\mathbb H}$. We find that they are both isomorphic to the Klein-four group $(\mathbb{Z}/2\mathbb{Z})^2$. Along the way, we classify all nondegenerately braided fusion categories over $\mathbb R$ that have Frobenius-Perron dimension 4, and we determine exactly which of these categories is a Drinfeld center. We end the paper by discussing a homotopy-theoretic conjecture that organizes all these structures.