On classification of Q-Fano 3-folds of Gorenstein index 2. II

In the previous paper, we obtained a list of numerical possibilities of Q-Fano 3-folds X with Pic X = Z(-2K(X)) and h(0)(-K-X) greater than or equal to 4 containing index 2 points P such that (X, P) similar or equal to ({xy + z(2) + u(a) = 0}/Z(2)(1,1,1, 0), o) for some a is an element of 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.