On the Mext groups of sVec_R and sVec_H

We compute the groups of minimal nondegenerate extensions of the real symmetric fusion categories ${\mathrm{s}\mathrm{V}\mathrm{e}\mathrm{c}}_{\mathbb{R}}$ and ${\mathrm{s}\mathrm{V}\mathrm{e}\mathrm{c}}_{\mathbb{H}}$. We find that they are both isomorphic to the Klein-four group $(\mathbb{Z}\mathrm{/}2\mathbb{Z}{)}^2$. 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.