They can if it's relevant. Generally a psychological examination is part of qualifying for law enforcement work, and that involves evaluating mental health in terms of past history as well.
I certainly wouldn't want a crazy person on the verge of a killing spree to be getting a job as a cop.
He couldn't handle it. He's not crazy, imo, he's just socially developmentally delayed like all aspies (cops have to be socially sharp obviously) and has some kind of learning disorder like most of them