The following is an exercise in Kurzweil and Stellmacher:
Let $x$ be an involution in a finite group $G$, $K$ be a component of $C_G(x)$ and $L$ be a component of $G$. Show that $K$ normalizes $L$.
Here, $E(G)$ is the layer of the group $G$.
My attempt: I first showed that either $K$ is a component of $G$ or $[K, C_{E(G)}(x)]=1$.
If $K$ is a component of $G$, then $K=L$ or $[K,L]=1$ and we're done.
In the other case, I showed that $gg^x\in C_{E(G)}(x)$ for all $g\in L$. Thus, $(LL^x)^k=LL^x$ for all $k\in K$.
But how to show that $L^k=L$ for all $k\in K$?
PS: My question is not a duplicate of this because there StefanH asks for the mistake in his (wrong) argument.