Rational quadric bundles and group quotients

Question

Suppose I have a rational projective variety $X$ and a quadric bundle $Q \to X$ such that the total space of $Q$ is rational. Assume now that I operate on $X$ with a finite group $G$ and that the quotient $X/G$ is still a rational projective variety. Assume that the quadric bundle structure is invariant w.r.t. the $G$-action on the base $X$ so it passes to the quotient giving a quadric bundle $Q/G \to X/G$. Is the total space of $Q/G$ rational?

Answer 1

What precisely do you mean by "quadric bundle"? If you mean a "family of quadric hypersurfaces", then that fails already for the plane conic bundle $Q/G$ over the projective plane $X/G$ associated to a pair of a smooth cubic threefold $Y\subset \mathbb{P}^4$ and a line $L$ in $Y$.

Let $X\to L$ be the projectivized normal bundle of $L$ in $Y$, i.e., an element of $X$ is a pair $(p,[\Pi])$ of a point $p\in L$ and a $2$-plane $\Pi$ containing $L$ such that the tangent space $T_p \Pi$ in $T_p\mathbb{P}^4$ is contained in $T_p Y$. Denote by $\rho:\mathbb{P}^4\setminus L \to \mathbb{P}^2$ a linear projection away from $L$. Then $\rho(\Pi \setminus L)$ is a point of $\mathbb{P}^2$. Thus, there is an induced morphism $\widetilde{\rho}:X\to \mathbb{P}^2$, $[\Pi]\mapsto \rho(\Pi\setminus L)$. This is a degree $2$, finite morphism. Denote by $G$ the group of automorphisms of this morphism, generated by an involution $\iota$ of $X$.

Let $Q$ be the parameter space of triples $(p,[\Pi],q)$ of a pair $(p,[\Pi])$ in $X$ and an element $q$ in the closure of $(\Pi\cap Y)\setminus L$. The projection $\text{pr}:Q\to X$ is a plane conic bundle. The involution $\iota$ lifts: for $\iota(p,[\Pi]) = (\iota(p),[\Pi])$, also $\iota'(p,[\Pi],q) = (\iota(p),[\Pi],q)$.

The quotient of $X$ by $G$ is the projective plane $\mathbb{P}^2$. Yet the quotient of $Q$ by $G$ is the blowing up of $Y$ along $L$. Since $Y$ is not rational, also $Q/G$ is not rational.