Almost subnormal subgroup

Definition
A subgroup $$H$$ of a group $$G$$ is termed an almost subnormal subgroup (or sometimes f-subnormal subgroup) of $$G$$ if there exists a chain $$H = H_0 \le H_1 \le H_2 \dots \le H_n = G$$, such that for $$0 \le i \le n - 1$$, $$H_i$$ is either a defining ingredient::normal subgroup or a defining ingredient::subgroup of finite index in $$H_{i + 1}$$.