Sub-(isomorph-normal characteristic) subgroup

Definition
A subgroup $$H$$ of a group $$G$$ is termed a sub-(isomorph-normal characteristic) subgroup if there exists an ascending chain of subgroups:

$$H = H_0 \le H_1 \le H_2 \le \dots \le H_n = G$$

such that each $$H_i$$ is an isomorph-normal characteristic subgroup of $$H_{i+1}$$, i.e., $$H_i$$ is characteristic in $$H_{i+1}$$, and $$H_i$$ is isomorph-normal in $$H_{i+1}$$: every subgroup of $$H_{i+1}$$ isomorphic to $$H_i$$ is normal.

Stronger properties

 * Weaker than::Isomorph-free subgroup
 * Weaker than::Isomorph-normal characteristic subgroup
 * Weaker than::Sub-isomorph-free subgroup

Weaker properties

 * Stronger than::Characteristic subgroup