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 $n=1$.