What is the ruling on a husband abandoning his wife by going to his family's home, and them inciting him to divorce, along with his betrayal of her, and is there a specific period after which the wife is no longer permissible for him?

It is obligatory for both spouses to treat each other with kindness. If the wife is disobedient (nashaz), the husband should admonish her, then shun her in bed (not outside the home), then, if admonition and shunning are ineffective, he may hit her with a non-bruising strike. There is no limit to shunning for a legitimate Shar'i reason. For a husband to shun his wife outside the home, or before admonition, or without any disobedience on her part, is an injustice and poor marital conduct. A husband's shunning does not render his wife unlawful to him, and she has the right to bring her case before a judge if he shuns her without a legitimate Shar'i reason. It is advised to communicate with the husband and understand the reasons for his shunning, and for the wife to remedy any shortcomings, or for wise individuals to intervene for reconciliation. If the husband insists on divorce, then divorce may be a good outcome. As for accusing the husband of betrayal (flirting with women), one must be cautious not to slander him without proof, because it is among the gravest sins. If it is proven that he committed indecency and has not repented, it is preferable for the wife to separate from him through divorce or khul'.