Birationally rigid varieties with a pencil of Fano double covers. II

The study of the birational geometry of Fano fibrations $\pi\colon V\to\mathbb P^1$ whose fibres are Fano double hypersurfaces of index 1 is continued. Birational rigidity is proved for the majority of families of this type, which do not satisfy the condition of sufficient twistedness over the base (in particular, this means that there exist no other structures of a fibration into rationally connected varieties) and the groups of birational self-maps are computed. The principal components of the method of maximal singularities are considerably improved, chiefly the techniques of counting multiplicities for fibrations $V/\mathbb P^1$ into Fano varieties over the line.