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

[Bleys]

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?

 

[Office_Shredder]

What if N=G?

 

[tmccullough]

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.