First, the standard disclaimer when it comes to medical issues... I am not a doctor and you should talk to your doctor before doing exercise with an injury.
Some of this is going to depend on the exact location and nature of the injury. The problem with injuries in the groin (especially upper groin) and abdominal area is that it's your core muscles. That means that almost any strenuous exercise is going to have the possibility of hurting or re-injuring because people tend to tighten that area when they strain other muscles. Even if you could immobilize the area, you might still tense the muscles. Given that, I would stay away from any heavy lifting, even upper body.
Si Philip's suggestion of using a hand cycle is excellent since it could be low stress and if you're seated properly shouldn't put any stress on the groin area.
I might also try a modified crunch that's even shorter than a regular crunch. When you do a sit-up from your back, the initial part of the motion uses your psoas muscles (which connect in at the groin area) rather extensively. As your body rises up you start to engage the abdominal muscles more and the psoas muscles less. If you started up near the top and just did short motions you might be able to almost completely eliminate the psoas muscles from the picture. I haven't tried this and again, I'm not a doctor, so you need to be extremely careful if you want to go this route. If you feel any groin pain AT ALL, stop immediately. If you have access to an ab machine (the kind that you can sit up in) and you can rig it to allow for the shorter motion at the end of the spectrum then that might work even better as you can get into position more easily without using the groin (with a crunch you still need to sit up for the first one). Just use very little or no weight to start.
The MOST IMPORTANT piece of advice though is this: being out of commission for some amount of time may stink, but it's much worse if you end up re-injuring the muscle and causing an even longer time injured. If you need to completely take time away from working out then do that and spend the time concentrating on other areas - improved diet, better sleep habits, mental health, etc. If you push things too soon it ends up being counterproductive because instead of the small improvement in health from one or two extra workouts you could end up causing a long, extended time injured, so it's just not worth the risk.