Federation ball game. 7th ining, 0/0 game. No body is on base. Batter has a 3 balls and 1 strike count. Pitcher delivers a pitch that the batter swings and misses and the catcher drops. The batter believing that it was a dropped third strike takes off to first base. Seeing this the catcher throws to first and the ball ends up in right field. The batter goes and stops at second with out a play made on her. Since the catcher throw to a fielder and not the pitcher with no one on base it would be considered ball four and the batter would be awarded first bases which she took and advanced to second. If the umpire determines that this was done unintentionally does the runner stay on second or come back to bat with full count? If the umpire believes that this was a coached play does that change the decision?

In my game I brought her back with full count and we just laughed at it a little. But by book should she not be on second?

I think I'd have done the same thing in your sit. Max. Hopefully it wouldn't happen in a tourney or championship type game where coaches and fans can't take a joke.

Nobody on base, no harm, no foul. Bring her back with a full count, remind her one plus one is two (not 3).
NOW if someone was on base, (2nr or 3rd) You have a VERY DIFFERENT STORY.