Any recent inverter worth it's salt has the capability to automatically start a generator. At least one that is equipped with electric start and remote control capabilities. Of course there are a number of pieces of criteria to be met. Things like battery voltage below a certain point, battery state of charge below a certain level, regularly scheduled generator exercise, etc. Depending on which inverter you have it can do a number of different things. It is capable of starting the generator, can let it warm up for a given period of time, run a complete charge cycle, and run cool down time before turning the generator off. So the question is, would this be a useful feature to take advantage of?
Personally, I have never liked the idea of letting a computer determine when to run my generator, potentially even when I am not at home to monitor things. Of course, this is one of the most attractive features to some...the ability to keep the batteries charged even while away from home. But for me, there is just too much that could go wrong and potentially malfunction while I am away. Moisture in fuel, diesel engine not starting during cold weather, or any of a myriad of potential problems could happen. And I have no need to use a generator to charge my batteries when I am away from home. That's what the solar panels are for!
Why you might want it?
If, however, you are on a generator-battery system without other renewable sources of charging, this might be more attractive to you. But for me, I leave the solar hooked up to the batteries and they are totally charged when I return from my trip. No need for a generator, even when I leave the refrigerator on. But to safely do that, you need to have experience that proves your solar will more than keep up with any potential loads while you are away, even during very cloudy weather. I don't leave much if anything on while I am on a trip, so I have no concerns about that.