Quotients of conic bundles

By Castelnuovo criterion each unirational surface is rational if the ground field is algebraically closed of characteristic 0. But it does not hold if the field is not algebraically closed. For example, any Del Pezzo surface of degree 4, 3 or 2 is unirational but not all of them are rational. One of the case of unirational surfaces is quotients of rational surfaces by finite group. Applying G-MMP we get that any quotient is birationally equivalent to quotient of a Del Pezzo surface or a conic bundle.
We study when quotients of conic bundle are rational. The main result is following.
Quotient of rational conic bundle is either rational or birational equivalent to a quotient of a conic bundle by a group $C_{2^k}$, $D_{2^k}$, $A_4$, $S_4$ or $A_5$.
For all groups listed above we construct example when the quotient is not rational. Moreover, the space of obtained nonrational quotients has infinite dimension.
As a corollary of this result we get that if $G$ is a finite group acting on a rational surface $X$, order of $G$ is odd and not equal to 3 then the quotient $X/G$ is rational.