What is the ruling on applying for scholarships in different countries, and is it impermissible to travel for such scholarships accompanied by a large group of female students and organizers?

It is permissible to apply for a scholarship if travel is with a mahram, and this is preferable. If that is not possible, and there is a need for study, and fitnah (temptation/strife) is secure, then it is permissible to act upon the opinion of those who allowed a woman to travel without a mahram if accompanied by a trustworthy group. It must be noted that any statement that is disrespectful towards Allah and His divine decrees is impermissible, for what Allah ordains is always best for His servant.