Equivariant intermediate Jacobians and intersections of two quadrics

We present a short proof of the following theorem of Hassett and Tschinkel: for every finite group $G$, a $G$-equivariant smooth complete intersection of two quadrics in ${\mathbb{P}}_{\mathbb{C}}^5$ is projectively $G$-linear if and only if it contains a $G$-invariant line.