Cyclotomic extensions with split Galois group $\newcommand{\Gal}{\mathrm{Gal}} \newcommand{\Q}{\mathbf Q}$
Consider the set of all Galois extensions $E/\Q(\zeta_n)$ of a given cyclotomic field $\Q(\zeta_n)$ such that 
$$
\Gal(E/\Q) \simeq\Gal(E/\Q(\zeta_n)) \rtimes \Gal(\Q(\zeta_n)/\Q).
$$
In other words, such that there is a homomorphism 
$$
\Gal(E/\Q) \leftarrow \Gal(\Q(\zeta_n)/\Q)
$$ 
inverting the natural quotient map 
$$
\Gal(E/\Q) \to \frac{\Gal(E/\Q) }{\Gal(E/\Q(\zeta_n))}\simeq \Gal(\Q(\zeta_n)/\Q).
$$
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"?
 A: The splitting of the Galois group of Hilbert class fields of an extension field is 
discussed in the following articles


*

*B. Wyman, Hilbert class fields and group extensions, Scripta math. 29 (1973), 141–149

*R. Gold, Hilbert class fields and split extensions, Ill. J. Math. 21 (1977), 66–69

*R. Bond, On the splitting of the Hilbert class field, J. Number Theory 42 (1992), 349–360

A: $\newcommand{\Gal}{\mathrm{Gal}} \newcommand{\Q}{\mathbf Q} \newcommand{\Z}{\mathbf Z} \newcommand{\F}{\mathbf F}$
Abbreviate $K=\Q(\zeta_n)$.  Note first that a galoisian extension $E$ of $K$ need not be galoisian over $\Q$, so I'm assuming that you are considering only those $E$ which are.  We then have an exact sequenece
$$
1\to\Gal(E|K)\to\Gal(E|\Q)\to\Gal(K|\Q)\to1
$$
in which the last group is $(\Z/n\Z)^\times$, of order $\varphi(n)$.  A sufficient condition for the sequence to split is : the degree $[E:K]$ is prime to $\varphi(n)$ (Schur-Zassenhaus).  I don't think there is a classification of all such extensions.
Note finally that this answer does not depend on the fact that $K$ is the cyclotomic field of level $n$, or even the fact that the base field is $\Q$.   It applies to any galoisian tower $E|K|F$: the associated short exact sequence
$$
1\to\Gal(E|K)\to\Gal(E|F)\to\Gal(K|F)\to1
$$
splits if the degrees $[E:K]$, $[K:F]$ are mutually prime.
Addendum (at Alex Bartel's suggestion): Let's return to the case $F=\Q$, $K=\Q(\zeta_n)$, $\Delta=\Gal(K|\Q)$, and suppose that $n$ is a prime $p$, for simplicity.  Kummer theory tells us that abelian extensions $E|K$ of exponent dividing $p$ correspond bijectively to subgroups $D\subset K^\times/K^{\times p}$ under $E=K(\root p\of D)$; such an $E$ is galoisian over $\Q$ if and only if the subgroup $D$ is $\Delta$-stable.  When such is the case,  we get examples of the kind of extensions envisaged in the question, with "split Galois group".  I guess the group $\Gal(E|\Q)$ will be commutative if and only if the $\Delta$-action on the $\F_p$-space $D$ is via the "mod $p$" cyclotomic character, namely the canonical isomorphism $\Delta\to\F_p^\times$. 
