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$\newcommand{\Gal}{\mathrm{Gal}} \newcommand{\Q}{\mathbf Q} \newcommand{\Z}{\mathbf Z}$Z} \newcommand{\F}{\mathbf F}$Abbreviate$K=\Q(\zeta_n)$. Note first that a galoisian extension$E$of$K$need not be galoisian over$\mathbf Q$, \Q$, 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 $\Delta\to{\mathbf F}_p^\times$. \Delta\to\F_p^\times$. 6 added 3 characters in body$\newcommand{\Gal}{\mathrm{Gal}} \newcommand{\Q}{\mathbf Q} \newcommand{\Z}{\mathbf Z}$Abbreviate$K=\Q(\zeta_n)$. Note first that a galoisian extension$E$of$K$need not be galoisian over$\mathbf 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$D$is via the "mod$p$" cyclotomic character$\Delta\to{\mathbf F}_p^\times$. 5 deleted 2 characters in body$\newcommand{\Gal}{\mathrm{Gal}} \newcommand{\Q}{\mathbf Q} \newcommand{\Z}{\mathbf Z}$Abbreviate$K=\Q(\zeta_n)$. Note first that a galoisian extension$E$of$K$need not be galoisian over$\mathbf Q$, 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$D\subset K^\times!/K^{\times K^\times/K^{\times p}$under$E=K(\root p\of D)$; such an$E$is galoisian over$\Q$if and only 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$D$is via the "mod$p$" cyclotomic character$\Delta\to{\mathbf F}_p^\times\$.