I had the same thing happen on my guest machine today. I saw the battery at 10V and then 7V when trying to start. Fuel pump relay kicking in and out like it was keeping time to a song..... that battery is 2 and 3/4 yrs old. Ordered some MotoBattery AGM batteries for both my machines as the other has to be right behind since replaced at the same time. I am sure that is the issue in my case. We will see how the MB AGM performs in the long run....... Anything over 3 yrs on lead acid battery and you are on borrowed time, IME.
If battery is showing 12.7 volts two things come to mind.
Try cranking / putting on a load. Still showing 12.7 volts? Start checking connections. Battery, alternator, grounds etc. All the usual suspects. You can have full voltage but a loose or corroded connection won't allow the amps necessary.
Try cranking, voltage drop after putting a load on it? Probably bad battery. Load test if possible before straight out replacing. But like @fortrdb said, if its older its probably time to replace.