Is NH Always a Normal Subgroup of G When N and H Are Normal Subgroups?

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There was an exercise in a book to prove that given N is a normal subgroup of a group G if H is also another normal subgroup of G the NH (the set of elements of the form nh for n in N and h in H) is a normal subgroup of G. That was all fine but I was wondering if the converse is true. Considering the exercise didn't ask to do this I'm guessing no, but I'm finding it hard to create a counter-example.
I'm trying to use the symmetric group. I thought that the alternating group A_{4} could maybe be constructed from the Klein 4 group and another non-normal subgroup but I don't know how to show this, or if it's even true. Are there maybe simpler examples?
 
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What if N=G?
 
If H is the klein-4 group, and N = <g> where g\in A4\setminus H, then the product NH=A4 just based on index.
 

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