$H{'}$ is a subgroup of the quotient group $G/N$ $\Leftrightarrow$ $H{'}=H/N$ for some $H$ such that $H$ is a subgroup of $G$ and $H\supset N$.

Question

I am afraid that my question is very stupid.

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Let $G$ be a group.
Let $N$ be a normal subgroup of $G$.

Then,

$H{'}$ is a subgroup of the quotient group $G/N$
$\Leftrightarrow$
$H{'}=H/N$ for some $H$ such that $H$ is a subgroup of $G$ and $H\supset N$.

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Proof:
($\Rightarrow$)
Let $H:=\{a\in G\mid aN\in H{'}\}$.
Let $X\in H^{'}$.
Then, $X=aN$ for some $a\in G$.
So, $a\in H$.

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Fact1:
Let $b\in H$.
Then the following holds:

$x\in\{x\in H\mid b^{-1}x\in N\}$.
$\Leftrightarrow$
$b^{-1}x=n$ for some $n\in N$ and $x\in H$.
$\Leftrightarrow\,\,\,\,$ (Note that $b\in H$ and $n\in N\subset H$)
$x=bn$ for some $n\in N$.
$\Leftrightarrow$
$x\in bN$.

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By Fact1, $X=aN=\{x\in H\mid a^{-1}x\in N\}\in H/N$.

Conversely, let $X\in H/N$.
Then, $X=\{x\in H\mid a^{-1}x\in N\}$ for some $a\in H$.
By Fact1, $X=aN$.
Since $a\in H$, $aN\in H{'}$.
So, $X\in H{'}$.

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($\Leftarrow$)
Suppose that $H{'}=H/N$ for some $H$ such that $H$ is a subgroup of $G$ and $H\supset N$.
We show $H{'}\subset G/N$ holds.
Let $X\in H{'}$.
Then, $X=\{x\in H\mid a^{-1}x\in N\}$ for some $a\in H$.
By Fact1, $X=aN$.
So, $X\in G/N$.

Since $H{'}$ itself is a group, $H{'}$ is a subgroup of the quotient group $G/N$.

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https://math.stackexchange.com/a/3278269/384082

When I read the above answer to a question, I feel uncomfortable.

I proved the following fact:

$H{'}$ is a subgroup of the quotient group $G/N$
$\Leftrightarrow$
$H{'}=H/N$ for some $H$ such that $H$ is a subgroup of $G$ and $H\supset N$.

$H{'}$ is a set whose elements are left cosets of $N$ in $\color{blue}{G}$.
$H/N$ is the set of the left cosets of $N$ in $\color{red}{H}$.
So, when I write $H{'}=H/N$, I feel uncomfortable.
I want to write $H{'}\cong H/N$.
Is it really ok to write $H{'}=H/N$?