Reporting a student who may be psychologically disturbed
The "administration" is probably the wrong place to look. Instead, see if you your institution has some mental health staff, whether just associated contacts, or counselors, or even a dedicated behavior concerns advice function, to get proper advice. If the university is well-organized enough to have one of the latter, chances are they've also established policies requiring you to pursue this. And such an advice line would be well suited to tell you how to go about it. If your university doesn't, figure out where to report it.

Note that, unless your institution is highly dysfunctional, reporting concerning behavior and having a professional talk to the student in question shouldn't be considered a punishment, but a safety net.

As Buffy points out in a comment, the law may further require you to do some things, and avoid doing other things. While one would hope that the mental health support staff are well-informed, contacting the legal staff (University Counsel) may be useful. That said, as a non-lawyer person, I imagine most legal pitfalls concern naming the student when you shouldn't. Your initial inquiries need not include the name of the student, until you're convinced that you should.