Let $G$ be a finite nonabelian group.

Let $F_2$ be the free group with generators $x,y$, then we know its outer automorphism group is isomorphic to $\text{GL}_2(\mathbb{Z}$). Let $\text{Aut}^+(F_2)$ be the (index 2) subgroup of $\text{Aut}(F_2)$ mapping onto $\text{SL}_2(\mathbb{Z})\subset\text{GL}_2(\mathbb{Z})$ (via some fixed isomorphism $\text{Out}(F_2)\cong\text{GL}_2(\mathbb{Z})$).

Let's say that $G$ is "nice" if there is a surjection $\rho : F_2\twoheadrightarrow G$ with the property that *every* surjection $F_2\twoheadrightarrow G$ can be written as $\alpha\circ\rho\circ\varphi$ where $\alpha\in\text{Aut}(G)$, and $\varphi\in\text{Aut}^+(F_2)$.

Say it's "almost nice" if we allow $\varphi\in\text{Aut}(F_2)$.

Is every 2-generated group "nice" or "almost nice"?

Is there some similar class of groups that people have studied? It's important for my research to have a nice characterization of these groups, however I'm not a group theorist and hopefully I won't have to start from scratch when analyzing these groups.

thanks