In A. Mann's paper: Enumerating finite groups and their defining relations (1998, J. group theory), that can be found Here ,

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!