Thanks for any comment or answer.
Suppose $G$ is a finite group. Then we call $H$ is a $p$-local subgroup of $G$ if $H=N_G(P)$ for some $p$-subgroup $P$ of $G$. My question is: Is it possible to characterize or have a theorem about group $G$ such that $G$ is not equal to union of all 2-local subgroups, i.e. $G\neq \cup N_G(P)$ where indexed over all 2-subgroup $P$ of $G$. It is not hard to see $S_3,D_{10}$ are simple example for such groups.
A little bit too long for a comment.... And I write in haste, so all of the details need to be checked....
It might be possible to prove some kind of reduction theorem here: if you look at $F^*(G)$, then you can assume that it has no normal $2$-group (or $G$ itself is 2-local and you're done).
So now consider $F(G)$, and let $S$ be a Sylow $t$-subgroup of $F(G)$ for some odd prime $t$. You can look at the quotient $G/C_G(S)$; if this quotient satisfies the criterion (it's a union of 2-locals), then the same is true of $G$. So this becomes a question about a group $L$ such that $F^*(L)$ is a $t$-group. We're really thinking about automorphisms of $p$-groups here...
On the other hand, consider the layer $E(G)$, and suppose that $T=T_1\circ \cdots T_k$ is a central product of isomorphic quasisimples that is normal in $G$. Again you look at the quotient of $G/C_G(T)$ -- if it satisfies the criterion, then the same is true of $G$. So this becomes a question about a group $M$ such that $F^*(M)$ is a direct product of isomorphic quasisimples. This is really a question about quasisimples, and CFSG might yield an answer.
The final part of the reduction would require that you show that if $G$ satisfies the criterion, then a quotient of one of the two types just described must satisfy the criterion. I think this follows because, by Bender's Theorem, the intersection of all the centralizers is just $Z(F^*(G))$ and, since we're assuming $F^*(G)$ has no normal $2$-groups, $Z(F^*(G))$ has odd order. Now if $G$ satisfies the criterion, then any quotients by groups of odd order clearly satisfy the criterion, and this can be used to complete the reduction...
