I went through this task and forgot about it, so the following is only my assumptions. Try talking to Karen before the first lesson, if Karen's first lesson. What actions did you do when you tried to run this job?
I'll try to list it.
-> Cannot talk to Karen before lesson (Panty inspection, the lack of it, then direct start of the lesson)
-> talking to Karen after the lesson only leads to the usual "Sorry, i didn't mean to bother you" answer from Kate
-> I relaunched the whole Extra Lesson event with Andrew several times. No changes
-> I went for special punishments at 11:30 and 16:00, no changes either
-> I tried both Home punishments events with and without Karen. Nope neither.
-> I ofc went back at the Principal Fuck summons in his office
-> I went for several Cheerleading pimp'n'stalk repeatables. Nothing more.
-> I went for both Swimming Class events to. No effect
-> Nothing triggers by waiting at home, either during the week, or the week-end. And nobody shows up at the door except Nick and his, well, needs.
-> No classmate has anything to say that triggers anything. I didn't try all teachers.
-> I tried to go to class with or without impending fees to purge by working for Zoe, just to see if it changed things to have a fee or not. Doesn't seem to have had any effect but I can't be sure I duplicated all that's written up here, in both fee amount states (0 versus ++amount).
I wouldn't say I was thorough but I just can't figure any idea that would be related.
In your previous answers, you talked about a quest supposedly called "Support for the Team", which I don't have and can't find in my quest log history (so I figured maybe it was the stage I need to unlock before going any further. Do you have such a quest in your log ? No hint in the Investigation quest log on your side, I guess)
Maybe it's just a random Karen talk that triggers after an amount of reiterations along with simply going to class, that hasn't showed up yet. After all, it's a regular feature within Glamour (which gave me headaches along the Julia & Nick questline XD). The thing is, each time Karen shows up she either directly begins class or inspect panties, then begins class, without giving any space to talk to her beforehand.
As for possible sources of bugs. There seems to be a conflict between the Extra Lesson with Andrew and the Special 11:30 punishment with Sophie.
In order to initiate the Andrew lesson, you have to quit the classroom after 11. Otherwise, the special punishment with Sophie triggers automatically. And even if you quit the room at 11 and get back at 11:30+, sometimes, the Andrew lesson doesn't trigger and special punishment triggers in its state. Plus, I don't even have to have been actually punished, or fined to trigger the 11:30 special punishment. So maybe it is this event that conflicts with the possible lesson. Or maybe it's the panty inspection that conflicts with triggering the talk with Karen I'd need. Maybe both.
I also seem not to be able to get caught during class tchating to Sophie normally. Can't be sure of that as it's random, but no hiccups after around ten attempts seem odd.
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