\({\mathcal F}\)-constraint of the automorphism group of a finite group (Q1061221)

If \({\mathcal F}\) is a homomorph and \({\mathcal F}'=\{G|\) \(S^{{\mathcal F}}=S\) \(\forall S\leq G\}\), a group G is \({\mathcal F}\)-constrained if there is a maximal normal \({\mathcal F}\)-subgroup \(\bar M\) of \(\bar G=G/G_{{\mathcal F}'}\), such that \(C_{\bar G}(M)\leq \bar M\). The following result of the second author is known: if \({\mathcal F}\) is a saturated Fitting formation and G a group verifying i) \(G_{{\mathcal F}'}\leq \Phi (G)\); (ii) \(G/G_{{\mathcal F}'}\) has no direct abelian factors; iii) G is \({\mathcal F}\)- constrained, then Aut G is \({\mathcal F}\)-constrained. In this paper, the authors prove the above result when \({\mathcal F}\) is a homomorph closed for direct products and normal subgroups that is: i) saturated or ii) closed for central extensions.