In Harvard Abstract Algebra lectures (video here, minute 50:06), professor Gross shows the groups $G=\mathbb{Z}\big /4\mathbb{Z}$ and $H=2\mathbb{Z}\big /4\mathbb{Z}$ as an example in which a subgroup $H$ of a group $G$ does not map isomorphically to the quotient group $G\big/H$.
However, in order to convince myself I've tried to find an isomorphism between the two hoping to fail, but I am seeing there is a map that to me satisfies the conditions for an isomorphism:
$$\varphi: H\longrightarrow G\big/H$$
where these groups can be more explicitly written $H=\{0+4\mathbb{Z},\enspace2+4\mathbb{Z}\}$ and $G\big/H=\{ \underbrace{(0+4\mathbb{Z}) + H}_{=\{ 0+4\mathbb{Z}, \enspace 2+4\mathbb{Z}\}},\enspace \underbrace{(1+4\mathbb{Z})+H}_{=\{ 1+4\mathbb{Z}, \enspace 3+4\mathbb{Z}\}}\}$. Which I define as the map that sends the elements: $$0+4\mathbb{Z}\longmapsto (0+4\mathbb{Z}) + H \\ 2+4\mathbb{Z}\longmapsto (1+4\mathbb{Z}) + H$$
and, which the explicit definition I give is bijective, and which I verify is homomorphic. For example:
$\varphi( \underbrace{\left(0+4\mathbb{Z}\right) + \left(2+4\mathbb{Z}\right)}_{=(0+2)+4\mathbb{Z}=2+4\mathbb{Z}}) = \varphi(2+4\mathbb{Z})=(1+4\mathbb{Z}) + H = \left( \left(0+4\mathbb{Z}\right) + \left(1+4\mathbb{Z}\right)\right)+H= \underbrace{\left(0+4\mathbb{Z}\right) + H}_{\varphi(0+4\mathbb{Z})} + \underbrace{\left(1+4\mathbb{Z}\right) + H}_{\varphi(2+4\mathbb{Z})}$
The professor claims that every of the two elements of $H$ map to $H$, that none go to the other coset.
Could someone please help me to clarify this?