I am watching Benedict Gross's abstract algebra lectures, and wanted to clarify with someone one comment he makes.
He defines the mapping $f: G \to \text{Aut $G$}$ for some group $G$ and proves that it is a homomorphism. He establishes, generally, that its kernel is the center of $G$. All of this makes sense to me. But, instead of defining the image of $f$ generally, he gives an example of the Klein four-group, whose image contains only the identity element.
My question is: is there a way to define the image of this mapping generally for any $G$? Does it depend on $G$, even though the kernel does not?
As @Arturo points out the image is the group of so-called inner automorphisms, which are defined by conjugation by elements of $G$. For some groups, like all symmetric groups except $S_2$ and $S_6$, it's the entire automorphism group.
The Klein four group is abelian, which explains the example you mentioned: the center is everything.