One possibility is her social skill (especially in terms of expressive language), is not as good as her other abilities. I guess that she finds it difficult to initiate the social relationship (e.g. as simple as saying "hello, I'm XXX" is too embarrassing for her already) and that's why she prefers staying with you or any person she is familiar with.
School might not be a problem but may be you can consider locating a social skill group with small group size for her. When I say small I mean less than 8.
親子班might not solve the problem as it doesn't focus on her weakness. It might even make the situation worst as your problem is not in parent-child relationship but HER social skills.