${\mathfrak{A}}_5$-equivariant geometry of quadric threefolds

We classify $G$-Mori fibre spaces equivariantly birational to smooth quadric threefolds with fixed-point free actions of the alternating group $G={\mathfrak{A}}_5$. We deduce that such quadric threefolds are $G$-solid and the $G$-actions on them are not projectively linearizable.