Find an upper bound for the nilpotent class in Hall's criterion On Derek J.S. Robinson's A Course in the Theory of Groups, Hall's Criterion is:

If $N$ is a normal subgroup of $G$, and $N$ and $G/N'$ are nilpotent, then $G$ is nilpotent.

There is an exercise:

Find an upper bound for the nilpotent class in Hall's criterion.

At first, I thought I was asked to find the upper bound for the nilpotent class of $G$ which satisfied the conditions ($N \lhd G$, $N$ and $G/N'$ being nilpotent). But this wouldn't be possible, because for any nilpotent group $G$, if we take $N$ to be $G$ itself, then the conditions are certainly satisfied. I think this might means giving an upper bound for the nilpotent class of $G$ as a function of $c$ and $d$ if the nilpotent classes of $N$ and $G/N'$ are $c$ and $d$ respectively. 
Would you please give me some help, hints or references?
Many thanks.
 A: The paper Hall (1958) shows that the class of G is bounded above by $$d\binom{c+1}{2}−\binom{c}{2}.$$  This was improved in Stewart (1966) to $$cd+(c−1)(d−1)$$ and shown to be sharp: for every pair $c,d$ of positive integers there is a group $G$ of class $cd+(c-1)(d-1)$ which has a normal subgroup $N$ of class $c$ such that $G/[N,N]$ is of class $d$.
The proof uses a lemma on commutator calculus: $$[[N,N], [G,G,\dots,G]] \leq \prod [[N,G,\dots,G],[N,G,\dots,G]]$$
where the number of $G$s in each factor on the right hand side of the containment is equal to the number of original $G$s. Stewart's estimate then follows by induction on $c$ and (what appears to be) a straightforward simplification of the class of the right hand side of the containment.
For my own sanity:
$$\left[ d\binom{c+1}{2}−\binom{c}{2} \right] - \left[ cd+(c-1)(d-1) \right] = (d-1)\binom{c-1}{2} \geq 0$$


*

*Hall, P.
"Some sufficient conditions for a group to be nilpotent."
Illinois J. Math. 2 (1958) 787–801.
MR105441
URL:euclid.ijm/1255448649

*Stewart, A. G. R.
"On the class of certain nilpotent groups."
Proc. Roy. Soc. Ser. A 292 (1966) 374–379.
MR197573
DOI:10.1098/rspa.1966.0140
JSTOR:2415662
