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We study two rational Fano three-folds with an action of the icosahedral group   $\mathfrak{A}_5$  . The first one is the famous Burkhardt quartic three-fold, and the second one is the double cover of the projective space branched in the Barth sextic surface. We prove that both of them are   $\mathfrak{A}_5$  -Fano varieties that are   $\mathfrak{A}_5$  -birationally superrigid. This gives two new embeddings of the group   $\mathfrak{A}_5$   into the space Cremona group.

Burkhardt Quartic, Barth Sextic, and the Icosahedron