Using a Galois connection to compute character degrees

Given a Mersenne prime $q$ and a positive even integer $e$, let $F$ and $E$ be the fields of orders $q$ and $q^e$ respectively. Let $C$ be a cyclic subgroup of $E^\times$ whose index in $E^\times$ is divisible only by primes dividing $q - 1$. We compute the character degrees of the group $C \rtimes {\rm Gal} (E/F)$ by using the Galois connection between the subfields of $E$ and the Galois group ${\rm Gal} (E/F)$.