Abelian fully invariant subgroup

Definition
A subgroup $$H$$ of a group $$G$$ is termed an abelian fully invariant subgroup or fully invariant abelian subgroup if $$H$$ is an abelian group as a group in its own right (or equivalently, is an abelian subgroup of $$G$$) and is also a fully invariant subgroup (or fully characteristic subgroup) of $$G$$, i.e., for any endomorphism $$\sigma$$ of $$G$$, we have $$\sigma(H) \subseteq H$$.

Examples based on subgroup-defining functions and series

 * For a solvable group, the penultimate member of the derived series (i.e., the last member before reaching the trivial subgroup) is an abelian fully invariant subgroup.
 * For a nilpotent group, second half of lower central series of nilpotent group comprises abelian groups: In particular, this means that for a group $$G$$ of nilpotency class $$c$$, all the subgroups $$\gamma_k(G), k \ge (c + 1)/2$$ are abelian characteristic subgroups.