It is well known that any group is a homomorphic image of a free group. I want to know more about this theorem when $G$ is a finite simple group. Does there exist any reference to state about it? Thanks in advance!
Based on the comment you added to your question, one relationship is that any group that is generated by $k$ elements is a homomorphic image of a free group of rank $k$. This is an immediate consequence of the universal property for free groups.