Equivariant rationality of Fano threefolds in the family \textnumero 2.12

We prove that a faithful group action on the smooth complete intersection $X$ of three divisors of bidegree $(1,1)$ in ${\text{p}}^3\times {\text{p}}^3$ is linearisable if and only if $\text{rk}({\text{pic}}^G(X))\mathrm{\ne }1$.