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

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 $\mathbb P^3$, a quadric $Q\subset\mathbb P^4$, 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.
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