Inverse Galois problem for non-Galois extensions

The inverse Galois problem asks whether every finite group appears as the Galois group of a Galois extension of the rational numbers.

Is anything known about the anologous problem, where the extensions are not required to be Galois? In other words, for a finite group $$G$$, does there exist a finite field extension $$K$$ of $$\mathbb{Q}$$ such that $$\mathrm{Aut}(K/\mathbb{Q})=G$$?

Is this suspected to be as difficult as the inverse Galois problem or easier?