Sub-weakly marginal subgroup

Definition
A subgroup $$H$$ of a group $$G$$ is termed a sub-weakly marginal subgroup if there exists an ascending chain:

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

such that each $$H_i$$ is a weakly marginal subgroup of $$H_{i+1}$$.