When is mapping $g$ to $g^{-1}$ a group homomorphism? When is mapping $g$ to $g^{-1}$ a group homomorphism? It just means that the map maps the identity to the identity and inverses to inverses. So does that mean it's only a homomorphism if the inverse is itself?

Well $\varphi(ab) = (ab)^{-1} = b^{-1}a^{-1}$ and $\varphi(a)\varphi(b) = a^{-1}b^{-1}$. When are they equal for any $a$ and $b$?
