For the past one week, I have been *trying to learn more* about automorphism groups of different groups. Very recently one of my friend asked this question to me:

- What is the automorphism group of $(\mathbb{Q}^{\ast},\times)$. In short, what is $\text{Aut}(\mathbb{Q}^{\ast})$?

I emailed couple of friends and got the answer as:

- $\text{Aut}(\mathbb{Q}^{\ast})$ is isomorphic to the automorphism group of a free abelian group of countable rank. In particular, it will contain $\text{GL}(n,\mathbb{Z})$ for all $n$.

My question would be :

Can we realize $\text{SL}_{n}(\mathbb{Z})$ to be the automorphism group of some group?

Are there groups which are which are "

*very difficult*" to be realized as the automorphism group of a certain group.

So suppose someone comes and asks me: Is $S_{3}$ or $\text{GL}_{2}(\mathbb{Z})$ the *automorphism group* of some group, then how can I answer the question? I am particularly interested in seeing how to think for a solution.

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