Free group on countable set is quotient-universal for finitely generated groups

Statement

Let $\Omega$ be a countable set, and let $F$ be the free group on $\Omega$. Then, every Finitely generated group (?) is isomorphic to some Quotient group (?) of $F$.

Related facts

• Finitary symmetric group on countable set is subgroup-universal for finite groups
• Free group on two generators is SQ-universal