Thanks for the suggestions folks.
She's still reading 12.4V without the charger and won't start. For kicks I put the bike on a charger this morning. With the charger, 16.4V. Bottom line, I don't think it's the battery as most battery problems I've encountered seem to start showing themselves in the 10V range. Plus, there's no drain in the battery when the starter is depressed so, for my money, I'm going with the starter solenoid. One last question: would a bad starter solenoid cause the headlight not to illuminate? Thanks.
I'll keep you posted!
