I asked the same question on stackexchange, but I didn't get an answer, so I am reposting it here in hope of one (or an appropriate reference to a textbook or otherwise). I am assuming all groups finite. Suppose $A$ is an elementary abelian $2$-group and $C$ is a cyclic group of odd order acting fixed-point-freely on $A$.
My question is: does this situation yield any structural information about their semi-direct product $\Gamma=A\rtimes C$?
For instance, $Z(\Gamma)=1$ obviously, since the action has no fixed points. Another immediate observation is that $\Gamma$ is solvable, but not nilpotent, since $C$ is self-normalizing. A more specific question: is it true in this situation that any two self-normalizing subgroups (of the same order) are conjugate? This last bit would follow at once from Carter's theorem if these subgroups were nilpotent, but that's not the case here.