In A. Mann's paper: Enumerating finite groups and their defining relations (1998, J. group theory), that can be found [Here](http://www.cecm.sfu.ca/~jborwein/ex2.pdf) , 


Mann's says (see pages 62-63):


" Let $F$ be a free group of rank d, and let $F\_i$ be the lower p-central series of $F$ . Let $ H= F/F\_{c+1} $ for some $c$ , and let $ G= H/N$ for some subgroup $N $ of $H\_c=F\_c/F\_{c+1} $ satisfying $|H\_c/N|=|N| $ or $ p|N| $ . ***Since H is the free group of rank d in the variety of groups with lower p-central series of length $c$ , two such factor groups $H/N,H/M$ are isomorphic if and only if $M$ and $N$ are conjugate under $Aut(H) $*** ".


Can someone help me understand the bold part ? Why is it true ? I understand that if we have an automorphism $\phi $ of $H$ , such that $\phi( N )= M $ , then the corresponding factor groups are isomorphic. But why in this case, if we have $H/M \equiv H/N$ i it implies that there is an automorphism of $H$ sending $N$ to $M$ ? 
Does someone have any good reference for this kind of theorem? Can someone help me understand it?
Does similar results hold for general free groups? (i.e- $F/N \equiv F/M $ iff $M$ , $N $ are conjugates under $Aut(F)$ ) Are there any similar results for other varieties (such as the variety of groups with p-derived series of length $c$ ? or even derived/central series of length $c$ ? 


Hope someone will be able to help me understand this


thanks everyone! 



Varieties Of Groups & Enumeration Of Size of Isomorphic Factor Groups
This property is not true in general for free groups. I have finally remembered a counterexample! There are 19 normal subgroups $N$ of the free group $F\_2$ of rank 2 with $F\_2/N \cong A\_5$. It was proved in


B.H. Neumann, and H. Neumann. "Zwei Klassen
charakteristischer Untergruppen und ihrer Faktorgruppen". Math. Nachrichten 4
(1951), 106-125


that there are two orbits of the action of ${\rm Aut}(F\_2)$ on these subgroups, with lengths 9 and 10.


As I said in my comment, in the situation in Mann's paper, we are considering normal subgroups $N$ of a free group $H$ of $p$-nilpotency class $c$, with $N \le H\_c$. For two such subgroups $M$ and $N$, an isomorphism $\psi:H/M \to H/N$ lifts, by freeness, to an endomorphism $\phi:H \to H$, which must satisfy ${\rm Im}(\phi)N = H$. Now any subgroup of a nilpotent group $H$ that projects onto $H/H'$ must be the whole of $H$, so $\phi$ must be an epimorphism and hence an isomorphism.


