Is it permissible for a woman to host her sister's son in her home in the presence of her young daughter and her two sons without her husband's knowledge?

A woman is not permitted to allow her sister's son into her husband's home if she knows or suspects that her husband dislikes it. The fundamental principle is that entering his home without his permission is forbidden. However, if the husband consents and there is no fear of temptation (fitnah), it is permissible for him to stay with her daughter, provided there is no relationship between them, and the daughter is in a separate room, maintaining her independence from the non-mahram (definition) in the use of the house's facilities, so that no mixing occurs. But if he is her mahram through a legally established suckling relationship, then he may sit with her, eat with her, and see what a man sees of his mahram women.