Sub-cofactorial automorphism-invariant subgroup

Definition
A subgroup $$H$$ of a group $$G$$ is termed a sub-cofactorial automorphism-invariant subgroup if there exists an ascending chain of subgroups:

$$H = H_0 \le H_1 \le \dots \le H_n = G$$

such that each $$H_i$$ is a cofactorial automorphism-invariant subgroup of $$G$$.

Stronger properties

 * Weaker than::Characteristic subgroup
 * Weaker than::Cofactorial automorphism-invariant subgroup

Weaker properties

 * Stronger than::Subgroup-cofactorial automorphism-invariant subgroup: Also related:
 * Stronger than::Left-transitively 2-subnormal subgroup
 * Stronger than::2-subnormal subgroup
 * Stronger than::Subnormal subgroup