Subgroups of Quotient Groups

Let G be a group and let [tex]N\trianglelefteq G [/tex], [tex]M\trianglelefteq G [/tex] be such that [tex]N \le M[/tex]. I would like to know if, in general, we can identify [tex]G/M[/tex] with a subgroup of [tex]G/N[/tex].

Of course the obvious way to proceed is to look for a homomorphism from [tex]G[/tex] to [tex]G/N[/tex] whose kernel is M, but I can't think of one.

What I actually want to show is a more specialized result (namely the case when finite [tex]G/N[/tex] is the nilpotent quotient of [tex]G[/tex] and [tex]G/M[/tex] is a maximal p-quotient of [tex]G[/tex] for some p dividing the order of [tex]G/N[/tex]) but the above is a lot cleaner and didn't yield obviously to a proof or counter-example so I thought I'd explore that first.
 

fresh_42

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I would look for a counterexample, although I found none as the small groups are all "too cyclic". We have
$$
M\rtimes G/M \cong G \cong N \rtimes G/N\text{ and } N \triangleleft M
$$
This means we have to look for the automorphisms and need a group, where ##G/M## operates differently on ##M## than ##G/N## does on ##N##. Therefore we need automorphism groups which have a few elements to select from.
 

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