TITLE: How does this strange phenomena happen in quotient of groups ?
QUESTION [9 upvotes]: 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 ?

REPLY [7 votes]: 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 $D_8/H_1\cong D_8/H_2$.