Finite Groups Admitting ap-Solvable Operator Group,

LetGbe a finite group andHbe an operator group ofG. Suppose thatHis π(G)-solvable. ThenGis solvable ifGhas a nilpotent maximalH-invariant subgroupM, withM2having no quotient group isomorphic toD8. We also prove thatGis solvable, ifHacts non-trivially onGbut acts trivially on all properH-invariant subgroups in an appropriately chosen class subgroups. The detailed structure of such a groupGis determined.