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her behavioral issues could be due to low blood sugar in the afternoon.
she has to eat lunch to keep her blood sugar up.
you should use your authority as an adult to encourage her to eat lunch daily.
you can put limits on her fun activities unless she eats lunch each day at school
you can also try to pack healthy foods that she will eat and enjoy for her to bring to school for lunch.
perhaps she doesn't like the food at school?
does she have to bring money to pay for food at school?