On Clifford theory with Galois action

Let $\widehat{G}$ be a finite group, $N$ a normal subgroup of $\widehat{G}$ and $θ\in \operatorname{Irr}N$. Let $\mathbb{F}$ be a subfield of the complex numbers and assume that the Galois orbit of $θ$ over $\mathbb{F}$ is invariant in $\widehat{G}$. We show that there is another triple $(\widehat{G}_1,N_1,θ_1)$ of the same form, such that the character theories of $\widehat{G}$ over $θ$ and of $\widehat{G}_1$ over $θ_1$ are essentially "the same" over the field $\mathbb{F}$ and such that the following holds: $\widehat{G}_1$ has a cyclic normal subgroup $C$ contained in $N_1$, such that $θ_1=λ^{N_1}$ for some linear character $λ$ of $C$, and such that $N_1/C$ is isomorphic to the (abelian) Galois group of the field extension $\mathbb{F}(λ)/\mathbb{F}(θ_1)$. More precisely, "the same" means that both triples yield the same element of the Brauer-Clifford group $\operatorname{BrCliff}(G,\mathbb{F}(θ))$ defined by A. Turull.

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