In the standard \( \mathcal{N} \) = (4, 4) AdS3/CFT2 with symN (T 4), as well as the \( \mathcal{N} \) = (2, 2) Datta-Eberhardt-Gaberdiel variant with symN (T4/ℤ2), supersymmetric index techniques have not been applied so far to the CFT states with target-space momentum or winding. We clarify that the difficulty lies in a central extension of the SUSY algebra in the momentum and winding sectors, analogous to the central extension on the Coulomb branch of 4d \( \mathcal{N} \) = 2 gauge theories. We define modified helicity-trace indices tailored to the momentum and winding sectors, and use them for microstate counting of the corresponding bulk black holes. In the \( \mathcal{N} \) = (4, 4) case we reproduce the microstate matching of Larsen and Martinec. In the \( \mathcal{N} \) = (2, 2) case we resolve a previous mismatch with the Bekenstein-Hawking formula encountered in the topologically trivial sector by going to certain winding sectors.
