Matrices for finite group representations that respect Galois automorphisms

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 \(\rho :H\rightarrow \textsf {GL} (n,L)\) 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 \(\rho ':H\rightarrow \textsf {GL} (n,L)\) 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 \(\textsf {ind} _{H,H\rtimes \langle \tau \rangle }(\rho )\) 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 \(\mathbb {Q}[\sqrt{5}]\) and a four dimensional irreducible representation of the double cover of \(A_7\) over \(\mathbb {Q}[\sqrt{-7}]\).