Equivariant geometry of odd-dimensional complete intersections of two quadrics

Fix a finite group G. We seek to classify varieties with G-action equivariantly birational to a representation of G on affine or projective space. Our focus is odd-dimensional smooth complete intersections of two quadrics, relating the equivariant rationality problem with analogous Diophantine questions over nonclosed fields. We explore how invariants – both classical cohomological invariants and recent symbol constructions – control rationality in some cases.