The Tambara-Yamagami (TY) fusion category symmetry \( \textrm{TY}\left(\mathbbm{A},\chi, \epsilon \right) \) describes the enhanced non-invertible self-duality symmetry of a 2-dim QFT under gauging a finite Abelian group \( \mathbbm{A} \) . We generalize the enhanced non-invertible symmetries by considering twisted gauging which allows stacking \( \mathbbm{A} \) -SPTs before and after the gauging. Such noninvertible symmetries can be obtained from invertible anyon permutation symmetries of the 3-dim SymTFT. Consider a finite group G formed by (un)twisted gaugings of \( \mathbbm{A} \) , a 2-dim QFT invariant under topological manipulations in G admits non-invertible G-ality defects. We study the classification and the physical implication of the G-ality defects using the SymTFT and the group-theoretical fusion categories, with three concrete examples. 1) Triality with \( \mathbbm{A} \) = ℤN × ℤN where N is coprime with 3. The classification was previously determined by Jordan and Larson where the data is similar to the TY fusion categories, and we determine the anomaly of these fusion categories. 2) p-ality with \( \mathbbm{A} \) = ℤp × ℤp where p is an odd prime. We consider two such categories \( {\mathcal{P}}_{\pm, m} \) which are distinguished by different choices of the symmetry fractionalization, a new data that does not appear in the TY classification, and show that they have distinct anomaly structures and spin selection rules. 3) S3-ality with \( \mathbbm{A} \) = ℤN × ℤN. We study their classification explicitly for N < 20 via SymTFT, and provide a group-theoretical construction for certain N. We find N = 5 is the minimal N to admit an S3-ality and N = 11 is the minimal N to admit a group-theoretical S3-ality.
