Also, consider scene structure. A scene is, in essence, a microcosm of story, with a beginning, middle and end, with rising conflict that comes to a head and is resolved (usually putting the protagonist in a different, often worse, state of mind or body). It starts with a pov character taking action (or deciding to take action or something along those lines, involving a goal and pursuit of it); the pov character struggles toward his/her goal, with the scene antagonist opposing that goal, and there's rising tension, and a resolution, where the goal is either achieved or blocked or some hybrid outcome. End of scene.
So, applying the elements of structure to pov, if the pov character has lost consciousness, the scene, logically, is over. The pov character can't be acting toward a goal, because he's unconscious. The pov character can't be involved in any conflict, because he's ... say it with me ... unconscious. The pov character can't even be reflecting on what to do next, because he's unconscious. If the conflict is resolved for the moment, the tension is gone, and the scene is over. The reader is ready to jump to a new conflict, a new scene. When you start the next scene, it will establish a transition from the moment of unconsciousness, plus establish a pov character (the same as before or a new one), with a goal and an action toward that goal, which is being opposed by someone until the struggle comes to some sort of resolution, and that scene is over.
If what matters in the scene -- the key conflict -- is what happens when Character A is unconscious, then it's probably not his scene -- he's not the one moving the story forward, the struggle/conflict is unrelated to him, even if he's the victim of it -- and would be more effective from another pov. If what happens while he's unconscious doesn't matter to the scene goal, then you can skip it, and, if necessary, mention it in passing later (e.g., he awoke in the hospital, and an orderly told him the cops found him in the alley, or another character tells him that, or you're in a new scene, with a new pov character finds him and struggles to get him to safety).
I'm trying to think of an example where it really matters what happens to an unconscious POV character while he's unconscious, where that's the focus of the scene, and I really can't. I'm sure someone could, but generally, what happens when the character is unconscious isn't important enough to show in real time, largely because it's inherently devoid of conflict.
JD