There exist subgroups of any given subnormal depth for any given tuple of nontrivial groups as quotient groups in a subnormal series

Statement
Suppose $$A_1, A_2, \dots, A_n$$ are nontrivial groups (some of them could be isomorphic to each other. Then, there exists a group $$G$$ with a subgroup $$H$$ such that $$H$$ has a subnormal series in $$G$$

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

such that each $$H_{i-1}$$ is normal in $$H_i$$ and $$H_i/H_{i-1} \cong A_i$$ and such that the subnormal depth of $$H$$ in $$G$$ is exactly $$n$$, i.e., $$H$$ has no shorter subnormal series in $$G$$.

Related facts

 * Normality is not transitive for any pair of nontrivial quotient groups: This is the $$n = 2$$ case.
 * There exist subgroups of arbitrarily large subnormal depth

Proof
For the proof, see the linked Math Overflow question page.