Your daughter is shy now but as she ages and has more experiences she may be more outgoing. I was a very shy girl. I did play with neighbor kids. I think that doing that was easier when I was young. Very few homes had a TV and we weren't plugged into iPods or cell phones. One of the kids in the neighborhood would initiate play and I would go along with it. I don't think I ever initiated play.
Since graduating college I have become very outgoing, talkative, and intiator of activity. I think that I made that change because of the encouragement I received over the years from my mother and an aunt. They never insisted that I do anything. They made suggestions and then let it drop. We were members of a church and thru attendance in classes, working in the nursery, and getting a general feeling of safety with people I grew to know helped a lot.
And so I would suggest that you get involved in social activity with your daughter. You go across the street and make friends with the mother. Hopefully you'll have something in common in addition to your daughters but if not the two of you share your concerns about parenting your daughters. After you've been there by yourself and found that the home has the simiar values and expections that you have for your daughter. Take your daughter with you to visit. Or arrange a playdate with the mother so that her daughter will come to your house for an hour. Or arrange to take the girls out for ice cream.
I think reasons that I learned how to be a friend and how to socialize is that I had other adults in church and in the neighborhood who were role models and who did arrange for me to play with their children. My mother was also quite shy and had no friends herself. She had 3 sisters and they were her social network. She knew the other families but mostly she didn't socialize with them even when their daughter had become my friend. She welcomed the children in her home and nearly always had cookies and milk for us until she became ill. If she would see a child playing in the yard she would tell me to come in for cookies and why don't you bring so and so with you. She knew the name because she had already met the parents.
I want to especially encourage you to not push your daughter into making friends. If she's not already there she's reaching the age during which she will not do things in order to show her independence. And this can be an unconcious act on her part. If she's already uncomfortable making friends you insisting that she make friends and frequently give her suggestions will cause her to withdraw even more. SHe may feel that she cannot do what you ask and this increases her feelings of inadequacies and cause her to withdraw even more.
Since she was shy herself my mother didn't concerned about my lack of friends. She treated me as being already complete and that it was OK to be shy. An aunt whom I infrequently saw, during a visit when I was 11 or 12 told me that she was concerned about my shyness and lack of friends and wanted to help. I no longer remember her words but I know that they were short and loving. She didn't tell me to do anything. She gave me an article cut from a magazine about how to make friends. That was the beginning of MY quest for learning how to make friends. It was all up to me. No one pushed me in any direction.
I don't remember much about the article but I do remember that one way to help yourself get involved is to think about the other person and what they might like. THat gave an opening for conversation and it took the attention off me. I was thinking of the other person.
And the article told me that many people are shy and afraid just as I was and to look for those people and get acquainted. My first friends were either shy or lacked confidence in some area of their life. We bolstered each other up.
A foster daughter came to live with me when she was 6. The first time I took her to day care she took a doll with her. She was shy, in a new home after many foster placements. The day care was downstairs and she met a child coming up the stairs and she handed her the doll and said "would you like to play with my doll?" I was so impressed with that.
I adopted her later and I now have a 7 yo grandchild who talks constantly. Definitely not shy. But she gets involved by doing a similar thing. She brings two stuffed animals and asks someone to play with her. Within a week several girls were bringing their stuffed animals to after school care and they were all telling a story with their animals.
Asking to ride bikes with her neighbor is a good way to get acquainted. But I think that you'll need to be the one to get it started. She will learn how to do it on her own from watching you and other adults. Kids learn a lot, especially socially, by watching role models.
If you are also shy and unsure, perhaps you could introduce her to an adult who is more outgoing and she could watch her.
I'm remembering that when my daughter was that age she played mostly with neighborhood girls and all of us mothers knew each other and would participate in activities with them. For example one mother would take the girls to a movie. Another would have a sleepover. That sort of thing and we mother's were the ones to arrange it. I would not let my daughter play with anyone if I didn't know the family at least well enough to know they held similar values in parenting as I did. THe mother's were the initiators and the protectors. In today's world most of us including children know that danger is out there. My grandaughter, when she was 4, started talking about kidnappers.
In summary I think I'm saying that your daughter needs someone to model behavior for her. Giving her books or articles on friendship and shyness will help if no lecture goes along with them. I learned a lot from American Girl magazine. I don't know if it's still published. Usually extreme shyness is a result of low self confidence. Doing things with her to build her confidence will help. Refrain as much as possible to say or do anything that makes her feel inadequate. Give her lots of praise for things well done. I learned after my daughter was an adult that she wanted me to say "I'm proud of you!" much more often. (like every week at least)
Also, I was rarely lonely as a child. I read, played out my own stories with my dolls, and had brothers who tormented me. :):):) I was somewhat lonely when we moved just before my senior year but I still had the support of my family. The loneliness started in earnest when I lived in a dorm at college. The loneliness pushed me into making friends. I believe that "necessity is the mother of invention."
Another idea is to talk with her teacher, probably the one from last year, and ask her opinion on how your daughter gets along at school. She may not be as shy there. And ask her for suggestions.
How long has your daughter staying in a group? And was it a group with a focus in which she was interested. I told my daughter that joining a group is a comittment and insisted that she stay in the group for one session or one season. Sometimes she complained but by the end she said she had enjoyed it, perhaps not greatly, but it was worth doing.
Seems I could write forever but I'll stop now.
Your daughter will be fine. She is whole and complete unto herself. Life is a matter of learning as you go along. And no one else can do the learning for someone else.