Assume we are given an extension of number fields or $\mathfrak{p}$-adic number fields $L/E/K$ where each extension is abelian and $L/K$ is only assumed Galois. Now take any element $\sigma\in \textrm{Gal}(E/K)$. We can extend $\sigma$ to an element of $\tilde{\sigma}\in \textrm{Gal}(L/K)$. If $\tilde{\eta}$ is another such extension, then $\tilde{\sigma}$ and $\tilde{\eta}$ are conjugate by an element of $\textrm{Gal}(L/E)$. It follows that $\textrm{Gal}(E/K)$ defines a well-defined action on $\textrm{Gal}(L/E)$ by conjugating with a lifting, since the result of the conjugation is independent of the particular lift.
My question is essentially the following: Will this action give us any useful information about the extension $L/K$? Does it fit into the functoriality of class field theory in some way? For example, taking $E/K$ to be quadratic and $L/E$ to be an abelian extension of degree $4$, then we have that the action is trivial if only if $L/K$ is also abelian, so it should let us distinguish between abelian extensions and non-abelian extensions. How far can this be pushed in general and how much information can we generally get out of it? Are there any other interesting actions induced by different groups appearing in class field theory?