R. Carter prooved that in finite soluble groups $G$ Carter subgroups $C$ exist and that they are conjugated. Furthermore they are exactly the nilpotent projectors: For every normal subgroup $N$ of $G$ the factor group $CN/N$ is maximal nilpotent in $G/N$ .
My question is whether in non-finite soluble groups Carter subgroups (if they exist) are nilpotent projectors.
They are maximal nilpotent also in the infinte case and also for non-solvable groups.
For my study I would like to know this for a semidirect product of a nilpotent nomral subgroup with an abelian subgroup.
(This question is on stackexchange, too, unanswered.)