Other formulation of the inverse Galois problem Is it right to say that the inverse Galois problem is equivalent to the following statement: 

Does every finite group $G$ occurs as a quotient of $\text{Gal}(\bar{\mathbb Q}/\mathbb Q)$?

I'm not sure if this is "quotient" or "subgroup" that I should write. Thank you for your clarifications.

You should write quotient: if $K$ is a finite Galois extension of $\mathbb{Q}$, then $\mathrm{Gal}(K/\mathbb{Q})$ is a quotient of $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ by the Fundamental Theorem of Galois theory.
