To add. If you are that shy about it. I would ask a group of friends not just the girl, so it looks different. Should be easy on you.
If the girl and you get alone than you two will be together and kind of alone even though others are there.
Next time, you have something to do, likewise it will be easier to ask the girl one by herself.
If you two do not get alone, you can always have friends to lean on. You will not feel that strange about.
Tips: Do not be loud and show off.
Remain calm and be a friend.
Do not talk about anything else except friendship.
Let her take it into dating, if not it might scare her.
Sometimes you can lead it there but I would be careful, because of mix singles people give.
For example: 3 yrs ago I thought this girl was blowing me off. I went with another girl and messed up. Now I just found out a few days ago the girl who I thought was blowing me off was the one I should have waited for. "Waiting on a woman"(Brad Presley) So true. Yes you can get mix up easy and think the other. Just watch that, because when you do, you loose good friends.
Another tip: Do not stare at her too much, a little glance long enough to smile and for her to see might be good.
I do strongly say ask her out and get 2 or 3 more friends. It's the best way, besides if she doesn't like you that way she will go because it's a group thing. She might not like you that now, but she might sooner or later. Remember be a friend. I have to repeat this a lot to people, because I have learn it was my flaws and others I know as well.
A girl who jumps in is most of the time something wrong with them. I learned that the hard way. I was married for 2 and half yrs and it was more like prison. Never again.