Definition (Property A). We will say that group $G$ has property A if for every finitely generated subgroup $H$ of $G$ there exists a group morphism $\rho : G \to F$ to a finite group such that $\rho(H) \neq \rho(G)$.
Of course, property A implies residual finiteness but seems to be a stronger property. (Edited later: As pointed out by YCor this is, of course, false! )
Property A is related to the concept of LERF groups but seems to be a weaker property. Recall that a group $G$ is called LERF if for every finitely generated subgroup $H$ of $G$ and every $g \in G \setminus H$ there exists a finite index normal subgroup $K$ containing $H$ but not containing $g$.
Question 1. Has property A already appeared in the literature ? Is it actually weaker then locally extended residual finiteness ?
My second question concerns finitely generated subgroups of matrices groups.
Question 2. Does there exist a finitely generated subgroup $G$ of $GL_n(\mathbb C)$ (for some $n$) which does not have property $A$ ?