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We prove that any G-del Pezzo threefold of degree 4, except for a one-parameter family and four distinguished cases, can be equivariantly reconstructed to the projective space , a quadric , a G-conic bundle or a del Pezzo fibration. We also show that one of these four distinguished varieties is birationally rigid with respect to an index 2 subgroup of its automorphism group. Bibliography: 15 titles.

Automorphisms of threefolds that can be represented as an intersection of two quadrics