Properties of $\operatorname{Aut}(X)$

Question

Let $X$ be a set and $\operatorname{Aut}(X)$ denote the group of all the automorphisms of $X$. If we know properties of $\operatorname{Aut}(X)$ then which properties of $X$ can be obtained if:

1. $X$ be a Group,
2. $X$ be a topological space,
3. $X$ be a topological manifold,
4. $X$ be a Lie group.

If $M$ be a topological manifold then which group actions (such as $\operatorname{Aut}(M)$) on $M$ is useful?

Thanks.

Answer 1

The following MO thread deals with the question:

when does the automorphism group of an object determine the object?

Answer 2

In all cases, the knowledge of $\operatorname{Aut}(X)$ can tell us something about $X$, but not too much. Let me illustrate this where $X=G$ is a group. For example, both $S_3$ and $C_2\times C_2$ have the same automorphism group $S_3$, but of course $S_3$ and $C_2\times C_2$ are quite different. On the other hand, if we know that $\operatorname{Aut}(G)$ is, say, trivial, then we know a lot about $G$, namely that $G$ is abelian and all elements $g$ satisfy $g^ 2=e$. In fact, $G\cong 1$ or $G\cong C_2$. There are many results of this nature.