For a finite group $G$ is there a subgroup $H$ such that for every chief factor $K/L$ of $G$ one has:


* $G = K C\_G(K/L)$ and $K \leq HL$ (so $K/L$ is inner and covered by $H$)
* $G \neq K C\_G(K/L)$ and $H \cap K \leq L$ (so $K/L$ is non-inner and avoided by $H$)


If $G$ is solvable, then all such $H$ are conjugate, called system normalizers, and are of the form $\cap N\_G(G\_{p'})$ where $G\_{p'}$ are Sylow $p$-complements.


A chief factor of a group is a pair of normal subgroups $L<K$ with $L,K \unlhd G$ such that if $L < M < K$ then $M$ is not normal.



Generalized system normalizer for insoluble finite groups
I think a nonsplit extension of a simple group by an outer automorphism would provide a counterexample.


The smallest example of this is the group of order 720 sometimes denoted $M\_{10}$, the extension $A\_6.2\_3$ in ATLAS notation. The chief factor $A\_6$ is not inner, whereas the one of order 2 is inner, so a subgroup $H$ as in the question would have to be a complement of $A\_6$. But the extension is nonsplit.


