Suppose that I have two normal subgroups $L, M$ in $H$ such that $L<M$ and $M / L$ is nilpotent of class $2$.
Suppose also that $[H,[H,H]]$ is contained in $[M,M]$.
Can I then conclude that $H / L$ is nilpotent? To be fair, I have no idea of how to prove it but I have never worked with nilpotent groups to have a counterexample neither.
I just know how to see that $H / L$ is nilpotent-by-nilpotent, since we have a short exact sequence $$1\to [H,[H,H]]L/ L \to H/L \to H/ [H,[H,H]]L \to 1,$$ but I do not know if the property $[H,[H,H]] < [M,M]$ gives us more information about $H/L$.
Thank you!