I’m learning the Fitting subgroup these days. I’m interested in this topic and particularly in the role that it plays in the structure of groups. Many people on MSE mentioned that the Fitting subgroup/generalized Fitting subgroup controls the structure of a group. Here are some quotes.

@Stephan mentioned in the comment that:

For a given Fitting subgroup $F$ there are just a finite number of subgroups $U$ which contain the center $Z(F)$, and also $\mbox{Aut}(F)$ is finite, and if $V \le \mbox{Aut}(F)$ also there are just a finite number of homomorphisms $\varphi : U \to \mbox{Aut}(V)$, so there are just a finite number of solvable groups that could be constructed as a semidirect product $G = V \ltimes_{\varphi} U$, in particular so that $U = C_G(F)$, in this way we have a bound on the number of groups that are possible. And that might mean "the Fitting subgroup controls the structure".

I can understand this. But as a beginner, I want to make sure this is a right way to think. So **my first question** is: Is his understanding correct?

@Geoff mentioned in the nice answer that ($E(G)$ below refers to the *layer* of a group):

The automorphism group of $E(G)$ has a normal subgroup $K$ consisting of the automorphisms which fix every component, and ${\rm Aut}(E(G))/K$ is a permutation group of degree $n,$ where $G$ has $n$ components. Also, $K/{\rm Inn}(G)$ is isomorphic to a subgroup of a direct product of outer automorphism groups of finite simple groups. Thus the structure of $F^{*}(G)$ controls the structure of $G$ to a large extent.

I got stuck here. He gave a good answer but I still got a few questions. He said $K$ is a subgroup of ${\rm Aut}(E(G))$, but he then used the notation “$K/E(G)$”. I wonder if $K/E(G)$ is well-defined since $K$ was not defined to contain $E(G)$. If not, it must be a typo, so here comes **my second question**: ~~What did he intend to refer to by “$K/E(G)$”?~~ (**EDIT:** I now know that it actually should be “$K/{\rm Inn}((E(G))$”) And could you explain why ${\rm Aut}(E(G))/K$ is permutation group of degree $n$ and why $K/{\rm Inn}(G)$ is isomorphic to a subgroup of a direct product of outer automorphism groups of finite simple groups? I also wonder how did the “**THUS**” come, namely how was it concluded that $\mathbf{F^*(G)}$ controls the structure of $G$ by giving some properties of $\mathbf{E(G)}$. I know it’s my problem and I know $F^*(G)=F(G)E(G)$.

**My third question**: Many people mentioned the outer automorphism group when talking about the structure of a group, but it seems to be quite a hard stuff to understand. What do I need to know about the outer automorphisms in terms of constructing a group?

Here are three questions and you can help me with commenting on or answering anyone of them. Or anything that you think can help me understand the importance of the Fitting subgroup in controlling the structure of a group is welcome. Any help is sincerely appreciated! Thanks!

THUSthe Fitting subgroup controls the structure of $G$”? I don’t see any relationship between them. $\endgroup$ – Hello May 1 at 14:01"in view of the above, considering the Fitting subgroup is important in understanding the structure of [finite] groups"and "thus" is just rhetorical. It's probably a bit more precise than that, but I'm not sure it's essential to figure out. $\endgroup$ – YCor May 4 at 14:05