Subgroups of Quotient Groups

[Hello Kitty]

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

Of course the obvious way to proceed is to look for a homomorphism from $$G$$ to $$G/N$$ 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 $$G/N$$ is the nilpotent quotient of $$G$$ and $$G/M$$ is a maximal p-quotient of $$G$$ for some p dividing the order of $$G/N$$) 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]

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.