Given a map of $n$-fold loop spaces $X\to Y$, we can take the homotopy cofiber, denote it $Y/X$ (all spaces here will also have a base point, and all maps pointed). I have some basic questions about this construction:

1. When is it the case that $X/Y$ is again an $n$-fold loop space?
2. Are there weaker conditions that guarantee $X/Y$ is an $m$-fold loop space for some $0<m<n$?
3. As a topological space, $X/Y$ satisfies a universal condition up to homotopy. In particular, it induces a morphism $X/Y\to Z$ whenever there is a morphism $X\to Z$ such that the composition $Y\to X\to Z$ is null. When is it true that this induced morphism is again a morphism of $n$- or $m$-fold loop spaces, assuming $X/Y$ is?

I feel like these should be some kind of "homotopy normality" conditions on $Y\subset X$, which I know that others have studied (e.g. Presma) for $n=1$.

(EDIT)

So, I don't want to delete the above, because I think it generated some useful stuff even thought it has some obvious mistakes and confusion in it.

But I want to clarify and try to correct my question. My question really is, I guess, the following: given a closed inclusion of Lie groups $H\to G$, the coset space $G/H$ is also the base space in a fibration $H\to G\to G/H$ (really a principal $H$-bundle). So in this situation, we have this sequence of spaces $H\to G\to G/H$ which satisfies (it seems to me) two universal properties. On one hand, it's a fibration, so in other words, $H$ is, up to homotopy, the fiber over the projection map $G\to G/H$. On the other hand, $G/H$ is precisely the coset space, so given a group homomorphism $\phi:G\to F$ such that $H$ goes to $1_F$, it would seem that one should obtain a factorization $G/H\to F$. Is this not true? In particular, though $G/H$ is not the homotopy cofiber of the map $H\to G$, it is $G/\sim$ for some equivalence relation, thus it admits such a universal property. More generally any continuous map $G\to Z$ such that any two points which differ by an element of $H$ go to the same place must factor through $G/H$ it seems.

Now really my questions is: how much of this relies on the property of being Lie groups? Can't we just do this for $E_n$-algebras in spaces? In general, is there some kind of homotopical Borel construction for a map of $E_n$-algebras which factors certain maps out of the "total space"?

And moreover, what conditions do we need on the inclusion of the sub-$E_n$-algebra to make sure that the quotient is still $E_n$ or $E_m$ for $m<n$?