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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. 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 $A \mapsto (\det A)A$ (this is an automorphism for all $n$ but they include it "for even $n$ only," which must mean they have a way of getting it from the rest for odd $n$ -- I haven't read the paper closely to see where that is indicated), and one additional automorphism when $n=2$.