Definition: A sequence $(G_0,G_1,\cdots,G_r)$ of subgroups of a group $G$ is called a normal series (or subnormal series) of $G$ if $$\{e\}=G_0\lhd G_1\lhd\cdots\lhd G_r=G$$ The factors of a normal series are the quotient groups $G_i/G_{i-1},1\leq i\leq r$.
If $G$ has a normal series $\{e\}=G_0\lhd G_1\lhd\cdots\lhd G_r=G$ then any subgroup $H$ also have one by using there exists a natural homomorphism, $$H_{i+1}\rightarrow G_{i+1}/G_i$$ Where $H_i:=G_i\cap H$ and further use $$H_{i+1}/H_i\cong G_{i+1}/G_i$$
I understand their role in order to prove this kind of theorems but can't think or visual that isomorphism (explicitly) by my own.
Any suggestions, comments, resources or solution will be appreciated.
updated: In section 3, Normal series under group constructions, there was similar argument for a subgroup, quotient group, and direct product.
Here, $G_i$ is normal in $G_{i+1}$
I don't see how to ensure an isomorphism from $H_{i+1}/H_i$ to A SUBGROUP of $G_{i+1}/G_i$. Like element of $H_{i+1}/H_i$ is $hH_i$ where $h\in H\cap G_{i+1}$, and element of $G_{i+1}/G_i$ is $gG_i$ where $g\in G_{i+1}$.