My suggestion is to let her join some interest groups.
When my daughter was about 1- 2 years old, she didn't like to interact with strangers and would cry loudly in the toddler groups. Don't give up.
Encourage her to interact with different adults and children can improve her self confidence (you may start with greeting or talking with the neighbours or safety guards)
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You can apply for insertion into kindergartens which can directly promoted into its primary schools (eg HK True Light in Tai Hang, SH, St Claires). Some of these primary schools are private schools which you can still enjoy "lucky draw".
It's much much easier to apply for insertion into kindergartens than applying for primary schools.