I want to consider the crossed module: $H \xrightarrow{t} Aut(H)$ for the case where $H = GL\_n(\mathbb{Z}) = Aut(T^n)$ is the automorphism group of the $n$-torus. Any suggestions on how to understand this automorphism group, $Aut(GL\_n(\mathbb{Z}))$, explicitly?



Automorphisms of $GL_n(\mathbb{Z})$
Hua and Reiner described this in their paper "Automorphisms of the unimodular group" (for them "unimodular" means determinant of absolute value 1, hence it's ${\rm GL}\_n(\mathbf Z)$ rather than just ${\rm SL}\_n(\mathbf Z)$) that appeared in 1951 [here](https://www.ams.org/journals/tran/1951-071-03/S0002-9947-1951-0043847-X/S0002-9947-1951-0043847-X.pdf). The end result (their Theorem 4) is that the automorphisms of ${\rm GL}\_n(\mathbf Z)$ are generated by inner automorphisms, the conjugate-transpose map $A \mapsto (A^{-1})^\top$, the map $A \mapsto (\det A)A$ (they include it "for even $n$ only," which must mean they have a way of getting it from the other generators for odd $n$ -- I haven't read the paper closely to see where that is indicated [Edit: see comment below]), and one additional automorphism when $n=2$. 


