Honda warranties the battery. Have them replace it if still under mileage.
If your FIT is loaded with electronics beyond the radio like Bluetooth, Nav and others they take a considerable draw on battery. I purchased a used MDX on the cheap I think because owner thought it had serious problems and was dumping it.
It went haywire a lot throwing all sorts of electrical fits, errant systems failures, check engine lights. I thought I purchased a bad car. Power window failure(needed a odd program). This is when it sat more than 5 minutes with key on. I called dealer when dash was lit up like christmas tree radio and nav locked out due to needing code over and over.
Dealer stated replace your battery first(that is what we do) and then come if in if persists. $100 was good to go. The car has been flawless since (80k-125k presently) except for failed AC relay ($7).