Is it permissible for the husband to prevent his wife from sponsoring her friend in a usurious bank to save her father from prison? What is the ruling of Islamic law and the husband's role in this case? And what if the wife provides the sponsorship without her husband's permission?

It is not permissible to guarantee someone who deals with a usurious bank, for that entails cooperation in sin. The husband must prevent his wife from this guarantee, because he is responsible for her according to the guardianship Allah has granted him, and he must obligate her to fulfill Allah's rights and restrain her from misdeeds.