Let $G=Gal(\bar{\mathbf{Q}}/\mathbf{Q})$ be the absolute Galois group over $\mathbf{Q}$.

Q1: Is it possible to find a (necessarily non-closed) normal subgroup $K\leq G$ such that $G/K$ is free of infinite rank ?

Q2: If the answer to Q1 is no then is it always possible to find a not necessarily continuous surjective homomorphism $\rho:G\rightarrow H$ where $H$ is an arbitrary finite group?

If you think that removing the continuity assumption does make the inverse Galois problem any simpler then please provide some explanations why.

Let $G=Gal(\bar{\mathbf{Q}}/\mathbf{Q})$ be the absolute Galois group over $\mathbf{Q}$.

Q1: Is it possible to find a (necessarily non-closed) normal subgroup $K\leq G$ such that $G/K$ is free of infinite rank ?

Q2: Let $H$ be a finite group. Is it always possible to find a non continuous surjective homomorphism $\rho:G\rightarrow H$?

If you think that removing the continuity assumption does make the inverse Galois problem any simpler then please provide some explanations.