How does this strange phenomena happen in quotient of groups ? in my question , here , i learned a strange fact from the comments which was a surprise for me on the answer of landsacpe  ! 
and this surprise it : 
if $G$ is a group , $H$ and $K$ are two normal subgroups of $G$ such that  $H$ is isomorphic to K then 
it is NOT necessary true that $G/H$ is isomorphic to $G/K$ . 
i always thought that two isomorphic groups have the same algebraic structure and so we can deal with them as the same things , we can use them respectively in fact ! 
but an example from the member Landscape broke down this thought 
so , why does this happens in quotients groups ? why two groups have the same isomorophism type leads to different  groups under quotients ? 
also , in which case , two isomorophism groups may leads to different things ? and in which case there is no matter to deal with them as the same thing ? 
till the moment , i think this phenomena may happens ONLY on infinite groups not finite groups " it may not happen also in infinite generated groups i think ! " 
any clarification plz ? 
 A: Some other examples that may help you:


*

*$G_1\cong G_2=\mathbb Z$, $H_1=2\mathbb Z\cong3\mathbb Z=H_2$ but $\mathbb Z_2=G/H_1\ncong G/H_2=\mathbb Z_3 $.

*$G_1\cong G_2=D_8$ and $H_1=\langle a\rangle\ncong H_2=\{e,a^2,b,a^2b\}$ but $D_8/H_1\cong D_8/H_2$.
A: The quotient of two groups is an operation on a group and one of its subgroups. The way that the subgroup sits inside the bigger group is relevant here. Looking just at the subgroup and replacing it with an isomorphic copy of it inside the bigger group may change the way that the smaller group sits inside the bigger one. As a simple example, consider $G=\mathbb Z_3\times S_3$. Now, consider the two subgroups $H=\mathbb Z_3\times \{e\}$ and $K=\{e\}\times \{e,(123),(132)\}$. Both are normal and isomorphic to each other. But $G/H\cong S_3$ while $G/K\cong \mathbb Z_3\times \mathbb Z_2$. 
In short, isomorphic objects can safely be interchanged (always with your eyes open though) inside the category where they are isomorphic. Once a construction is occurring outside of that category you can't just treat things that are isomorphic in some category as the same. When the category changes, the notion of isomorphism changes. In the case above, if you consider the category whose objects are $(G,H)$ where $G$ is a group and $H$ is a normal subgroup of $G$, and arrows $(G,H)\to (G',H')$ are group homomorphisms $\psi :G\to G'$ such that the restriction of $\psi$ to $H$ lands in $H'$, then isomorphic objects here will have isomorphic quotients. The quotient operation here becomes a functor from this category $C$ to the category $Grp$ of groups. 
In this category, for $(G,H)$ and $(G',H')$ to be isomorphic means that there is a group isomorphism $\psi:G\to G'$ such that the restriction of it to $H$ is an isomorphism onto $H'$. This says exactly that the isomorphism does not just change the names of the elements of the subgroups, but also the names of the elements of the bigger groups, and in such a way that everything is consistent. The counter example I gave above shows that $(\mathbb Z_3\times S_3, \mathbb Z_3\times \{e\})$ is not isomorphic to $(\mathbb Z_3\times S_3,\{e\}\times \{e,(123),(132)\})$.
