Is it permissible for a wife to spend the night at home with her female cousins for study purposes, with the knowledge of their families and hers, but without her husband's knowledge or permission? And is it permissible even if the husband agrees?

If a wife leaves and spends the night outside the home without her husband's permission, this is impermissible, as she must seek his permission merely to leave. However, if she allows women to spend the night in her home without her husband's permission, this is not forbidden to her unless she knows or strongly believes that he dislikes it. The preferred interpretation of "that they should not allow anyone whom you dislike to tread upon your فرش [bedding/furnishings]" is that it includes not permitting any disliked person, whether male or female or a close relative (mahram), to enter the home and sit in it, unless the husband knows or it is presumed that he does not dislike it.