Does intentionally swallowing vomit, or that which has come out of a fasting person's stomach and returned to it, break the fast, and is there any opinion in the four schools of thought that permits this?

Intentionally swallowing vomit or its equivalent after being able to expel it invalidates the fast. Invalidating an obligatory fast is not permissible; rather, one must preserve its validity and avoid its nullifiers. Scholars have differed on the ruling of vomit that returns on its own, without the fasting person's action:

Hanafis: If vomit returns on its own, it does not invalidate the fast according to Muhammad (which is the sound opinion among them), but it does invalidate it according to Abu Yusuf. However, if the fasting person returns it or if it returns in the amount of a chickpea or more, the fast is invalidated by their agreement. Malikis: Vomit breaks the fast, whether it returns intentionally or unintentionally, and qada' (making up the fast) is obligatory. Hanbalis: If vomit returns on its own, it does not invalidate the fast. However, if the fasting person returns it, then it invalidates the fast. Shafi'is: The return of vomit absolutely invalidates the fast, whether it is a little or a lot. The soundest opinion is that self-induced vomiting itself invalidates the fast, or the return of anything that came out, even if little.

It is clear that there is no opinion stating the validity of the fast of one who intentionally returns vomit.