Why might the inspector suspect Higgins and Pickering of “some improper purpose?” How does Mrs. Higgins respond to this? What is her point? (Act 5)

The inspector is suspicious of the two men's intent. Why are they searching for a young, unmarried girl? Mrs Higgins replies, "Well, of course he did. What right have you to go to the police and give the girl’s name as if she were a thief, or a lost umbrella, or something? Really!"

She responds this way because the men are acting as if they've lost a puppy...... or belonging. She's never seen Eliza as property or an experiment, and this have been one of her problems with the men's work all along.