Near-fields and linear transformations of finite fields

A question about near-fields suggests the following problem: If F is a finite field, K is a finite extension of F, and H is a multiplicative subgroup of K¹, describe the F-linear maps φ: K → K which fix F and leave each coset of H invariant. A plausible conjecture would seem to be that φ must be a field automorphism. This is confirmed here in the case that |H| and |F| satisfy a certain numerical relation (and, in particular, when K/F is quadratic). The bulk of the argument consists of showing that an F-linear transformation of K which preserves F-conjugacy is almost always an automorphism.