Show $\operatorname{Aut}(C_2 \times C_2)$ is isomorphic to $S_3=D_6$

Notice that $C_2\times C_2$ is a vector space over the field $Z_2$. Hence $Aut(Z_2\times Z_2)=Gl(2,2)$. It is easy to see that $Gl(2,2)\cong S_3 $.

There are $3$ non-trivial elements of $C_2 \times C_2$. Show that an automorphism must permute them and that any such permutation does in fact give rise to an automorphism. The group $D_6$ does the same thing, except to vertices of a triangle. So you can imagine writing the $3$ non-trivial elements on the vertices of a triangle to get the isomorphism explicitly !