On Clifford theory with Galois action

Abstract: Let $\widehat{G}$ be a finite group, $N$ a normal subgroup of $\widehat{G}$ and $\theta\in \operatorname{Irr}N$. Let $\mathbb{F}$ be a subfield of the complex numbers and assume that the Galois orbit of $\theta$ over $\mathbb{F}$ is invariant in $\widehat{G}$. We show that there is another triple $(\widehat{G}_1,N_1,\theta_1)$ of the same form, such that the character theories of $\widehat{G}$ over $\theta$ and of $\widehat{G}_1$ over $\theta_1$ 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 $\theta_1=\lambda<sup>{N_1}$ for some linear character $\lambda$ of $C$, and such that $N_1/C$ is isomorphic to the (abelian) Galois group of the field extension $\mathbb{F}(\lambda)/\mathbb{F}(\theta_1)$. More precisely, "the same" means that both triples yield the same element of the Brauer-Clifford group $\operatorname{BrCliff}(G,\mathbb{F}(\theta))$ defined by A. Turull.