From my past experience with Cap. One, they don't report the amount of your limit, so they may not report authorized user accounts either. Call them and ask. Sears should report. Usually takes 30-60 days.
I just added an AU to my CapitalOne card and on the agreement it does say that they will report on that persons credit but I wouldnt be able to tell you how long it will take before they actually do start to report.