Is it permissible to give an aunt's phone number to inquire about mixed-gender universities, in order to avoid severing kinship ties, or is that considered assisting in wrongdoing?

Giving the aunt's number to the mother is not aiding in wrongdoing; rather, it is aiding in good deeds, righteousness, and maintaining ties of kinship. Even if the brothers discuss mixed universities with the aunt, this does not make the act a forbidden aid to wrongdoing. Studying in mixed universities is not absolutely forbidden; it may be permissible when there is a need, provided the limits of Islamic law are observed.