AbstractIn the previous paper, we obtained a list of numerical possibilities of ℚ-Fano 3-folds X with Pic X = ℤ(−2KX) and h0(−KX) ≥ 4 containing index 2 points P such that (X, P) ≃ ({xy + z2 + ua = 0}/ℤ2(1, 1, 1, 0), o) for some a ∈ ℕ. Moreover we showed that such an X is birational to a simpler Mori fiber space. In this paper, we prove their existence except for a few cases by constructing a Mori fiber space with desired properties and reconstructing X from it.