Billie Bradley on Lighthouse Island by Janet D. Wheeler

One of Billie’s friends owned a summer bungalow on Lighthouse Island, near the coast. The school girls made up a party and visited the Island. There was a storm and a wreck, and three little children were washed ashore. They could tell nothing of themselves, and Billie and her chums set to work to solve the mystery of their identity. (From original advertisement) This is the third book in the “Billie Bradley” series, a mystery series for girls.