I have recently come across the result that ${\rm Aut}(\mathbb{Z}^n, +)$ is isomorphic to $GL_{n}(\mathbb{Z})$. Since transposition does not alter the determinant of a matrix, $A^{T} \in GL_{n}(\mathbb{Z})$ iff $A \in GL_{n}(\mathbb{Z})$.
This got me wondering if I transpose a matrix in $GL_{n}(\mathbb{Z})$, what happens to the corresponding automorphism in ${\rm Aut}(\mathbb{Z}^n, +)$?
Is there an easy way to describe what happens to said automorphism or does it perhaps depend on what isomorphism between ${\rm Aut}(\mathbb{Z}^n, +)$ and $GL_{n}(\mathbb{Z})$ we use?
Finally, on a related note, I would like to know is there a definition of transposition of a matrix that does not make reference to the elements of said matrix?