I am trying to align the following facts.

• The free group with two generators $F_{2}$ is isomorphic to a congruence subgroup $\Gamma(2)\le\mathrm{SL}_{2}(\mathbb{Z})$.
• The outer automorphism group of $F_{2}$ is isomorphic to $\mathrm{GL}_{2}(\mathbb{Z})$.

Surely the action of $\mathrm{Out}(F_{2})=\mathrm{GL}_{2}(\mathbb{Z})$ on $F_{2}=\Gamma(2)$ is not via conjugation. What is this action? Is there a neat, direct description of this action?

Added: It might be a long shot but what ultimately I wanted to ask was if there is a nice description of this action without really referring to $F_{2}$.