$G$ is a group $,H \cong K$, then is $G/H \cong G/ K$?
$G$ is a group with subgroups $H$ and $K$ such that $,H \cong K$, then is $G/H \cong G/ K$?
No. Consider $G = (\mathbb{Z},+)$, $H= (2\mathbb{Z},+)$ and $K= (4\mathbb{Z},+)$. Note that $H$ and $K$ are isomorphic by the mapping $z \to 2z$.
You might be interested in seeing the following questions as well.
Finite group with isomorphic normal subgroups and non-isomorphic quotients?
https://mathoverflow.net/questions/29006/counterexamples-in-algebra/