  We study two rational Fano threefolds with an action of the icosahedral group
$\mathfrak{A}_5$. The first one is the famous Burkhardt quartic threefold, 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.
