Nondegenerate extensions of near-group braided fusion categories

This is a study of weakly integral braided fusion categories with elementary rules to determine which possess nondegenerately extensions theoretically minimal dimension, or equivalently in this case, satisfy the modular extension conjecture. We classify near-group satisfying conjecture; remaining Tambara-Yamagami provide arbitrarily large families identical violating These examples generalize representation extraspecial $p$-groups for any prime $p$, only if they arise as adjoint subcategory twisted double an $p$-group.