The carpet should not extend - it ends where it should, at an intersection of vertical and horizontal. If you extended it, there would not be as defined an end as what you have now, and you would have to create some sort of threshold, which would would be awkward. The vinyl floor is not part of the stairs.

Lars

The stair and second floor are all covered by the carpet. In other words, behind door it is an another room with the carpet flooring. It is not part of the outside of the door. Thus, the vinyl floor should end at the doorway. Is it correct or I misunderstand something in flooring?

Do it how you would like. I have seen it done both ways, but as the previous poster indicates, extending the carpet to a location under the door will create an awkward and unnecessary transition. But, if that is what you think will look best...then do it that way.