A normal subgroup $N$ of $H$, such that $H$ is a subgroup of $G$, such that neither $N$ nor $H$ are normal in $G$. I am looking for an example of finite groups $N\lhd H\not\lhd G$ (but $H\leq G$), such that $N\not\lhd G$, and $H/N$ is non-trivial cyclic.
An answer to this should produce an answer to this (bountied) question.
 A: You can use GAP to find such examples easily. The following code will return such groups $G,H,N$ with $|G| \leq \texttt{max}$ and $|H/N| \geq \texttt{minquosize}$.
findSmallGroups := function(max,minquosize)
    for m in [1..max] do
        for n in [1..NrSmallGroups(m)] do
            G := SmallGroup(m,n);
            for ccH in ConjugacyClassesSubgroups(G) do 
                H := Representative(ccH);
                if not IsNormal(G,H) then
                    for ccN in ConjugacyClassesSubgroups(H) do
                        N := Representative(ccN);
                        if IsNormal(H,N) and not IsNormal(G,N) and IsCyclic(H/N) and Size(H/N) >= minquosize then
                            Print("result found for SmallGroup(",m,",",n,")\n");
                            return [G,H,N];
                        fi;
                     od;
                fi;
            od;
        od;
    od;
end;

The result obtained is the dihedral group $D_{12}$
gap> findSmallGroups(30,2);
result found for SmallGroup(12,4)
[ <pc group of size 12 with 3 generators>, Group([ f1, f2 ]), Group([ f1 ]) ]
gap> StructureDescription(last[1]);
"D12"

A: Easiest example is any non-abelian simple $G$, such as $G=A_5$, $H=A_4$, $N=\langle (1,2)(3,4),(1,3)(2,4)\rangle$
Smallest example I think is $G=S_3\times C_2$, $H=\{1,(1,2)\}\times C_2$ and $N=\{1,(1,2)\}\times\{1\}$
