What is the religious ruling regarding a wife who leaves her home without her husband's permission and lies to him and her family about her whereabouts and her son's whereabouts? And what should the husband do concerning his statement to her, "You are divorced"?

The wife erred by leaving without her husband's permission and by lying to him. As for the husband's utterance of divorce, what he said is considered an explicit divorce and thus takes effect. If this was not the third divorce, he has the right to take her back while she is in her iddah (definition) (waiting period), and it is recommended to have witnesses for this. If the iddah ends, then a new contract and a new dower (mahr) are required, provided they meet the conditions.