I would like to ask if there exists an explicit description of $\mathrm{Aut}(G)$, the group of automorphisms of a finite group $G$, in particular, when $G$ is abelian. E.g., if $G = \mathbb{Z}/m\mathbb{Z} \times \mathbb{Z}/m\mathbb{Z}$, where $m$ is a positive integer, how can we describe $\mathrm{Aut}(G)$? Which relation do we have between it and $GL_m(\mathbb{Z})$? If $m$ is prime then $\mathrm{Aut}(G) \cong GL_m(\mathbb{Z})$, but what happens for $m$ general? E.g., if $m=4$, I find that the cardinality of $\mathrm{Aut}(\mathbb{Z}/4\mathbb{Z} \times \mathbb{Z}/4\mathbb{Z})$ is $8$, by seeing to where the generators $(0,1)$ and $(1,0)$ are sent. But I have the feeling that $GL_4(\mathbb{Z})$ should be contained in $\mathrm{Aut}(\mathbb{Z}/4\mathbb{Z} \times \mathbb{Z}/4\mathbb{Z})$, so I should have a problem with the cardinalities.

More general, is it sufficient to see where a minimal set of generators is sent? Maybe someone could indicate me a paper where I could find a good description of such automorphism groups?

Automorphisms of direct products of finite groups, Arch. Math. (Basel) 86 (2006) no. 6, 481-489; and Bidwell, J.N.S.Automorphisms of direct products of finite groups II. Arch. Math. (Basel) 91 (2008) no. 2, 111-121. $\endgroup$ – Arturo Magidin Oct 16 '13 at 4:34