$\newcommand{\Gal}{\mathrm{Gal}} \newcommand{\Q}{\mathbf Q}$
Consider the set of all Galois extensions $E/\mathbb{Q}(\zeta_n)$$E/\Q(\zeta_n)$ of a given cyclotomic field $\mathbb{Q}(\zeta_n)$$\Q(\zeta_n)$ such that $$\mathrm{Gal}(E/\mathbb{Q}) \simeq \mathrm{Gal}(E/\mathbb{Q}(\zeta_n)) \rtimes \mathrm{Gal}(\mathbb{Q}(\zeta_n)/\mathbb Q).$$
$$
\Gal(E/\Q) \simeq\Gal(E/\Q(\zeta_n)) \rtimes \Gal(\Q(\zeta_n)/\Q).
$$
In other words, such that there is a homomorphism
$$\mathrm{Gal}(E/\mathbb{Q}) \leftarrow \mathrm{Gal}(\mathbb{Q}(\zeta_n)/\mathbb Q)$$ inverting$$
\Gal(E/\Q) \leftarrow \Gal(\Q(\zeta_n)/\Q)
$$
inverting the natural quotient map
$$\mathrm{Gal}(E/\mathbb{Q}) \to \frac{\mathrm{Gal}(E/\mathbb{Q}) }{\mathrm{Gal}(E/\mathbb{Q}(\zeta_n))}\simeq \mathrm{Gal}(\mathbb{Q}(\zeta_n)/\mathbb Q).$$$$
\Gal(E/\Q) \to \frac{\Gal(E/\Q) }{\Gal(E/\Q(\zeta_n))}\simeq \Gal(\Q(\zeta_n)/\Q).
$$
Are they classified? Is there a "largest" one? What can be said about them (or about their cohomology) in general? Are there any prominent examples of such extensions arising "in nature"?
