I've never flown a GPS approach with an autopilot, but I had a student who asked me this question:
You're flying a standard "T" approach, approach has been activated in the GPS, and you're approaching the final approach course at a right angle. Will the autopilot turn you 90 degrees onto the final approach course?
Like many things in aviation, "it depends."
If your autopilot has GPSS (GPS Steering), it will anticipate the turn and make a beautiful standard-rate turn that puts you smack on the final approach course.
If your autopilot simply has nav mode, it may either fly all the way to the point of the T, way overshoot final, and maybe make it back... I know before we added the GPSS to ours, by the time it realized what was going on, it had completely overshot final and gone full deflection so it'd just get confused and make a very slow turn.
Also, knowing how the GPSS works is essential. On the S-Tec, you put the autopilot in *Heading* mode, not Nav mode. Then, you flip the GPSS to "GPSS" and it feeds headings to the autopilot (as opposed to "HDG" which passes through the input from the DG bug). Every so often, someone will squawk the GPSS as not working, and every time when I go test it, it works fine. The problem is they're putting the autopilot in Nav mode, so even though the little GPSS light is on, the autopilot is ignoring its commands.
If your student owns a plane, s/he should have the autopilot manual. If it says nothing about GPS, chances are it doesn't have GPSS. Otherwise, many autopilot manuals are available for download on the internet.
Hope this helps!