Quotients of pointless del Pezzo surfaces of degree 8

In the talk we will consider del Pezzo surfaces of degree 8 over algebraically nonclosed fields of characteristic 0. Any quadric surface in three-dimensional projective space is a del Pezzo surface of degree 8, and it is well known that such surface can be pointless. We want to study birational classification of quotients of pointless del Pezzo surfaces of degree 8 by finite automorphism groups. In particular, we want to find conditions on the surface and the group for which the quotient can be not rational over the main field. We will show that the quotient by any group of odd order is birationally equivalent to the original surface, and the quotient by any group of even order is birationally equivalent to a quadric surface.