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?