I showed using a somewhat lengthy but elementary computation that the automorphism group $\operatorname{Aut}(D_{\infty})$ of the infinite dihedral group $D_{\infty} = \mathbb Z_2 * \mathbb Z_2$ splits as a semidirect product as follows: $$ \operatorname{Aut}(D_{\infty}) \cong \operatorname{Inn}(D_{\infty}) \rtimes \mathbb Z_2 $$ Here $\operatorname{Inn}(G)$ denotes the group of inner automorphisms of a group $G$. The $\mathbb Z_2$ factor in the above decomposition comes from the group homomorphism that interchanges the factors of $\mathbb Z_2 * \mathbb Z_2$. Since $D_{\infty}$ has trivial center, it follows that $\operatorname{Inn}(D_{\infty}) \cong D_{\infty}$. In particular we see that the group of outer automorphisms is given by $\operatorname{Out}(D_{\infty}) \cong \mathbb Z_2$.
What would be an approach to derive these results using some more theory (which might also be applicable to compute automorphism groups of more difficult groups)?
