0MY8 Proposition 3.12. There is a natural isomorphism E≅SW⋉ℚ[π0(G)].E\cong S^{W}\ltimes\mathbb{Q}[\pi_{0}(G)].
0MY9 Proof. If GG is connected this is (3.7). Otherwise the statements follow from a transfer argument for finite group torsors, see [SVW18, Section A.2.2]. ∎