$\def\QQ{\mathbb{Q}}\def\Gal{\text{Gal}}$Let $S_n$ act on $\QQ(x_1, x_2,\ldots, x_n)$. The field of invariants is generated by the elementary symmetric polynomials $\QQ(e_1, e_2, \ldots, e_n)$, so $\QQ(x_1, \ldots, x_n)/\QQ(e_1, \ldots, e_n)$ is Galois with Galois group $S_n$.
I would like an example of an extension $L$ of $\QQ(x_1, \ldots, x_n)$ such that
$L$ is Galois over $\QQ(e_1, \ldots, e_n)$ and
The surjection $\Gal(L/\QQ(e_1, \ldots, e_n)) \to \Gal(\QQ(x_1, \ldots, x_n)/\QQ(e_1, \ldots, e_n)) \cong S_n$ does not have a right inverse. In other words, I want $\Gal(L/\QQ(e_1, \ldots, e_n))$ not to be the semidirect product of $\Gal(L/\QQ(x_1, \ldots, x_n))$ by $S_n$.
If I replace $\QQ$ by $\mathbb{C}$, this is not too bad. Take $n=2$ and take the splitting field of $z^4 = e_1^2 - 4 e_2$, also known as $z^4 = (x_1-x_2)^2$. Then the cyclic group of $K$ over $\mathbb{C}(e_1, e_2)$ is cyclic of order $4$, coming from $z \mapsto i^k z$ for $k \in \mathbb{Z}/4 \mathbb{Z}$. But this doesn't work over $\mathbb{Q}$: If I take the splitting field of $z^4 = e_1^2 - 4 e_2$ over $\QQ(e_1, e_2)$, I get a dihedral group of order $8$ and I believe the extension is semidirect.
It is also straightforward to give examples where the $x_i$ are specific algebraic numbers, rather than formal variables. For example, $\QQ(\cos (\pi/8)) \supset \QQ(\sqrt{2}, - \sqrt{2}) \supset \QQ$ corresponds to $0 \to C_2 \to C_4 \to C_2 \to 0$.
But I am blanking on an example of the form $L/\QQ(x_1, \ldots, x_n)/\QQ(e_1, \ldots, e_n)$.
Motivation: In about a month, I am going to be proving that, in a tower $L/K/F$ with $L/F$ and $L/K$ splitting fields, every automorphism of $K/F$ extends to an automorphism of $L/F$. I want to point out that one needs to be careful; one can't claim that the action of $\Gal(K/F)$ on $K$ extends to $L$, so I need examples where $\Gal(L/F) \to \Gal(K/F)$ has no right inverse.
The two examples above are good ones, and are the ones I'll actually use in my teaching. But, when I set out to write the notes, I was reading "Galois Theory for Beginners", which always talks about $\QQ(x_1, \ldots, x_n)/\QQ(e_1, \ldots, e_n)$, so I started thinking about it, and now I have nerd-sniped myself.