There are three settings on the ISS mode selector...Solo, where only your handle will operate only your seat, Both, where if one handle is pulled, both seats go (rear first then about 1/3 of a second the front) and then Command Forward where the rear seat operates just itself but the front activates both (usually used for flying around non-pilots in the back seat). There's zero way the student could have pulled the handle and just had the instructor go. But, in my 13 years as a T-6 instructor pilot we flew around in Solo only for about 5 years due to concerns that the seat may not fire correctly or sequence correctly in the other modes.
If I had to speculate, it was an accidental activation. G-suit or other piece of gear caught in the handle with it not pinned. I know I've forgotten to pin the seat a few times after flights being tired/fatigue, in a hurry, etc. Hell, I've even dropped the pin while trying to safe the handle. It happens, even to someone with over a decade of experience. Certainly can happen to a new guy/FAIP.