Fitting subgroup is strictly characteristic

Statement
The Fitting subgroup of a group (i.e., the subgroup generated by all nilpotent normal subgroups) is a strictly characteristic subgroup: any surjective endomorphism of the whole group sends it to within itself.

Related facts
Similar results include:


 * Solvable core is strictly characteristic
 * Perfect core is fully characteristic

Facts used

 * 1) uses::Fitting subgroup is weakly normal-homomorph-containing
 * 2) uses::Weakly normal-homomorph-containing implies strictly characteristic

Proof
The proof follows directly by piecing together facts (1) and (2).