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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.

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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.

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  • $\begingroup$ Thanks. A6 is neither complemented nor Frattini, so I probably need to think about that case more carefully. $\endgroup$ – Jack Schmidt Jul 12 '13 at 21:19

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