Matrices for finite group representations that respect Galois automorphisms

Abstract: We are given a finite group $H$, an automorphism $\tau$ of $H$ of order $r$, a Galois extension $L/K$ of fields of characteristic zero with cyclic Galois group $\langle\sigma\rangle$ of order $r$, and an absolutely irreducible representation $\rho\colon H\to\operatorname{\sf GL}(n,L)$ such that the action of $\tau$ on the character of $\rho$ is the same as the action of $\sigma$. Then the following are equivalent. $\bullet$ $\rho$ is equivalent to a representation $\rho'\colon H\to\operatorname{\sf GL}(n,L)$ such that the action of $\sigma$ on the entries of the matrices corresponds to the action of $\tau$ on $H$, and $\bullet$ the induced representation $\operatorname{\sf ind}_{H,H\rtimes\langle\tau\rangle}(\rho)$ has Schur index one; that is, it is similar to a representation over $K$. As examples, we discuss a three dimensional irreducible representation of $A_5$ over $\mathbb{Q}[\sqrt5]$ and a four dimensional irreducible representation of the double cover of $A_7$ over $\mathbb{Q}[\sqrt{-7}]$.