Let \(G\) be a connected, simply connected, simple linear algebraic group over \(\mathbb{C}\), with diagram automorphism \(\omega\). The author wishes to study the analogue of the Borel-Weil-Bott theorem for the semi-direct product \(\langle\omega\rangle\ltimes G\). Choose an \(\omega\) invariant Borel subgroup and consider a one-dimensional representation \(\mathbb{C}_{\lambda,\zeta}\) of \(\langle\omega\rangle\ltimes B\). Here \(\zeta\) is the root of unity that tells how \(\omega\) acts and \(\lambda\) is a character of \(B\), necessarily fixed by \(\omega\). One has an \(\langle\omega\rangle\ltimes G\)-equivariant line bundle \(\mathcal L\) on \(G/B\) whose fiber at \(B\) is \(\mathbb{C}_{\lambda,\zeta}\). The author determines the action of \(\omega\) on the relevant cohomology group \(H^i(G/B,\mathcal L)\). Namely, if \(v\in W\) is the Weyl group element associated with \(\lambda\), then \(\omega\) acts on the highest weight space of \(H^{\ell(v)}(G/B,\mathcal L)\) by the scalar \((-1)^{\ell(v)-\widehat\ell(v)}\zeta\), where \(\ell\) is the length function on \(W\) and \(\widehat\ell\) its analogue for the fixed point subgroup \(W^\omega\). To see this the author imitates the proof in [M. F. Atiyah and R. Bott, Ann. Math. (2) 88, 451-491 (1968; Zbl 0167.21703)], combining a Lefschetz type fixed point formula with a Kostant type twining character formula.