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As I suggested in my short comment, this kind of question has been around for a long time and has led to a vast amount of literature. It probably starts with work over fields by Schreier and van der Waerden in the 1920s, then considerable work by Dieudonne, O'Meara, and many others. Two indications about what's out there are a survey by O'Meara The integral classical groups and their automorphisms (1969) and a short paper by me dealing from the algebraic group viewpoint with groups like $SL_n(\mathbb{Q})$, On the automorphisms of infinite Chevalley groups, Canad. J. Math. 21 1969 908–911, the latter probably available online by now.

These are listed on MathSciNet, but I couldn't display the links there for some reason. I'll try to suggest more focused literature on your question when I get time today.

P.S. Ed Formanek has pointed to a basic early paper by Hua and Reiner here. Like many other papers on related automorphism groups, the emphasis is on identification of special types of automorphisms which suffice to generate the whole group: inner, "field" or "ring" (as in complex conjugation and the like), "graph" (as in transpose-inverse map for special linear groups), "diagonal" (as in the use of conjugation by diagonal matrices not of determinant 1 to produce automorphisms of a subgroup). Naturally the ring or field automorphisms play no role over the rational integers or rational numbers.

2 added 589 characters in body

As I suggested in my short comment, this kind of question has been around for a long time and has led to a vast amount of literature. It probably starts with work over fields by Schreier and van der Waerden in the 1920s, then considerable work by Dieudonne, O'Meara, and many others. Two indications about what's out there are a survey by O'Meara The integral classical groups and their automorphisms (1969) and a short paper by me dealing from the algebraic group viewpoint with groups like $SL_n(\mathbb{Q})$, On the automorphisms of infinite Chevalley groups, Canad. J. Math. 21 1969 908–911, the latter probably available online by now.

These are listed on MathSciNet, but I couldn't display the links there for some reason. I'll try to suggest more focused literature on your question when I get time today.

P.S. Ed Formanek has pointed to a basic early paper. Like many other papers on related automorphism groups, the emphasis is on identification of special types of automorphisms which suffice to generate the whole group: inner, "field" or "ring" (as in complex conjugation and the like), "graph" (as in transpose-inverse map for special linear groups), "diagonal" (as in the use of conjugation by diagonal matrices not of determinant 1 to produce automorphisms of a subgroup). Naturally the ring or field automorphisms play no role over the rational integers or rational numbers.

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As I suggested in my short comment, this kind of question has been around for a long time and has led to a vast amount of literature. It probably starts with work over fields by Schreier and van der Waerden in the 1920s, then considerable work by Dieudonne, O'Meara, and many others. Two indications about what's out there are a survey by O'Meara The integral classical groups and their automorphisms (1969) and a short paper by me dealing from the algebraic group viewpoint with groups like $SL_n(\mathbb{Q})$, On the automorphisms of infinite Chevalley groups, Canad. J. Math. 21 1969 908–911, the latter probably available online by now.

These are listed on MathSciNet, but I couldn't display the links there for some reason. I'll try to suggest more focused literature on your question when I get time today.