Is it permissible to reserve the inheritance of a psychologically ill divorced sister—who supports four children—to buy an apartment in her name and her brother's name, to ensure she does not squander the money?

You are not permitted to place an interdiction on your sister due to her mental illness except through a Sharia judge. If the judge rules to interdict her and appoints you as a guardian over her wealth, then you may not dispose of it except in a way that serves her best interest. There is no objection to you partnering with her to purchase an apartment with both your funds if it is in her best interest; otherwise, it is not permissible. The wealth of an orphan, a spendthrift, and an insane person are treated the same, and it is not permissible to dispose of it except in a way that benefits the interdicted person.