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?

First observe that for any group $G$, the map $\phi$ defined by $\phi(g)  = g^{-1}$ is an injective surjection:  injective since $\phi(a) = \phi(b)$ implies $a^{-1} = b^{-1}$, hence $a = b$; a surjection since $\phi(a^{-1}) = (a^{-1})^{-1} = a$.  Furthermore $\phi(ab) = (ab)^{-1} = b^{-1}a^{-1} = \phi(b) \phi(a)$.
If $G$ is abelian, then $\phi(b) \phi(a) = \phi(a)\phi(b)$ so that $\phi(ab) = \phi(b) \phi(a) = \phi(a)\phi(b)$, and $\phi$ is an isomorphism.  Similarly, if $\phi$ is an isomorphism, then $\phi(ab)  = \phi(b) \phi(a) = \phi(ba)$, whence the injectivity of $\phi$ forces $ab = ba$: $\phi$ is an isomorphism if and only if $G$ is abelian.  QED.
Hope this helps.  Cheers, 
and as always,
Fiat Lux!!!
