Writing projective representations over subfields

Let G=〈X〉 be an absolutely irreducible subgroup of GL(d,K), and let F be a proper subfield of the finite field K. We present a practical algorithm to decide constructively whether or not G is conjugate to a subgroup of GL(d,F).K×, where K× denotes the centre of GL(d,K). If the derived group of G also acts absolutely irreducibly, then the algorithm is Las Vegas and costs O(|X|d3+d2log|F|) arithmetic operations in K. This work forms part of a recognition project based on Aschbacher's classification of maximal subgroups of GL(d,K).