We introduce a twisted fiber-bundle construction of quantum CSS codes over group algebras R=F₂[G], where each base generator carries a generator-dependent R-linear fiber twist satisfying a flatness condition. This construction extends the untwisted lifted product code, recovered when all twists are identities. We show that invertible twists (satisfying a flatness condition) give a complex chain-isomorphic to the untwisted one, so the resulting binary CSS codes have the same blocklength n and encoded dimension k. In contrast, singular chain-compatible twists can lower boundary ranks and increase the number of logical qubits. Examples over R=F₂[D₃] show that the twisted fiber bundle code can outperform the corresponding untwisted lifted-product code in k while keeping the same n and, in our examples, the same minimum distance d.