For the easiest case, assume that $L/E$ is Galois and $E/K$ is Galois. Under what conditions can we conclude that $L/K$ is Galois? I guess the general case can be a bit tricky, but are there some "sufficiently general" cases that are interesting and for which the question can be answered?
EDIT: Since Jyrki's reply seems to suggest that there is no general criterion on the groups. Can we say something if we put criterions on the fields? Assume say that $K=\mathbb{Q}$ or $K=\mathbb{Q}_p$?