Affine representability results in A^1-homotopy theory III: finite fields and complements

We give a streamlined proof of ${\mathbb{A}}^1$-representability for $G$-torsors under “isotropic” reductive groups, extending previous results in this sequence of papers to finite fields. We then analyze a collection of group homomorphisms that yield fiber sequences in ${\mathbb{A}}^1$-homotopy theory, and identify the final examples of motivic spheres that arise as homogeneous spaces for reductive groups.