I have read evidence that cosets $ uN $, or $ N $ is the core of some homomorphism, partitioning a group. However, it seems that only the fibers of any homomorphism seem to share a group.

Say a little homomorphism $ varphi: G rightarrow H $ maps $ u $, $ v $, $ p $, and $ q $ to exactly one element of the coded domain $ H $: $ U $, $ V $, $ P $, and $ P $ respectively (otherwise it is not a function). $ UP = VP $ it is only when $ U = V $. Since homomorphism only brings domain values to exactly one value in the coding domain, $ uP = vP $ it is only when $ u $ and $ v $ are in the same homomorphism fiber, so the fibers are shared $ G $. I said in the same fiber because also note that in this proof all the fiber elements above $ F $ is not even necessarily a subgroup of $ G $.

Is the proof correct?