In this talk, we study a question of Colliot-Thélène and Iyer concerning the existence of rational sections in families of homogeneous spaces over an abelian variety, after base change by a suitable étale isogeny of the abelian variety. Assuming characteristic zero and that the homogeneous spaces arise from connected reductive groups whose root datum contains no factor of type $E8$, we give a positive answer to the question posed by Colliot-Thélène and Iyer. Our approach relies on cohomological invariants such as the Milnor invariant and the Rost invariant. This leads us to a closer analysis of the action of the multiplication-by-n map on the unramified cohomology of abelian varieties, as well as on their motives with integral coefficients.