Find Aut$(G)$, Inn$(G)$ and $\dfrac{\text{Aut}(G)}{\text{Inn}(G)}$ for $G = \mathbb{Z}_2 \times \mathbb{Z}_2$ Find Aut$(G)$, Inn$(G)$ and $\dfrac{\text{Aut}(G)}{\text{Inn}(G)}$ for $G = \mathbb{Z}_2 \times \mathbb{Z}_2$. Here is what I have here: Aut$(G)$ consists of 6 bijective functions, which maps $G$ to itself, since Aut$(\mathbb{Z}_2 \times \mathbb{Z}_2) \approx S_3$. I think the next part goes wrong. For Inn$(G)$, letting $\kappa_x : G \rightarrow G$ to be the conjugate function, I found $\\{e, \kappa_{(0,1)}, \kappa_{(1,0)}, \kappa_{(1,1)}\\}$. I can't determine $\dfrac{\text{Aut}(G)}{\text{Inn}(G)}$ yet since I need to correctly determine Inn$(G)$. Any advices or comments?

First note that $G$ is abelian, therefore $Z(G) = G$
($Z$ is the Center of $G$).

Then you can use a little proposition (Humphreys pg.73-74) that tells you $\text{Inn}(G) \approx \frac{G}{\text{Z}(G)}$

The proof of this prop in a few words:

define an automorphism (prove it) $\varphi_x(g)=xgx^{-1}$, then define $\phi : G \to \text{Aut}(G)$ by $x \mapsto \varphi_x$ and find its image and its kernel, then apply first theorem of homomorphism and you conclude.