What is the ruling on the action of a husband who invites his brother's wife and her children to lunch at a restaurant without his own wife being present, considering this an obligation towards her, while the wife considers this action a betrayal due to her absence?

It is permissible for your husband to invite his brother's wife and her children in the presence of his sister, provided that the Sharia-compliant controls are observed, such as the hijab, lowering the gaze, avoiding seclusion (khalwa), and not speaking in a soft or alluring manner. However, if there is a Sharia violation or temptation involved, it is not permissible, and you should advise him gently. It is not the wife's right to discipline her husband or compel him to fulfill obligations, except in the context of enjoining good and forbidding evil. She must obey him in what is good, respect him, uphold his rights, and advise him gently. It is not permissible for a woman to travel without a mahram.