You have a hole in your story. If the group is super secret, no one else knows of it and they aren't telling anyone else, how would the sensei know of its existence?
Perhaps the loyalty of the super secret group isn't as strong as its members would like to believe it is. And what would compel individuals to question their own loyalty and abandon their oath of super secrecy to their private, super secret group to alert the sensei to a problem? Did they have a hazing ritual for potential members that some found offensive and similar to junior high immaturity? Or perhaps some members got far too catty and ugly about the other members of the club who were excluded or kicked out of the super secret group which left a sour taste in the more sensible members? The latter would create a terrible dilemma for some people who perhaps joined thinking it was just another fun offshoot of the original club but found they were caught up in a hateful pack mentality which, if they said anything, exposed them to ridicule or banishment. Did the behavior of the super secret group spill over into the main club in any way?
Why would the sensei asking a few questions to verify that the stories are indeed accurate be considered a bad thing? If the super secret group has done nothing to wrong others in the main club, why does the sensei's attempt to get to the truth distress them? I'd think they would want to exonerate themselves and make sure sensei knew they had been maligned with a false report by disgruntled "friends" out to make trouble for them.
The sensei has the right to protect his/her own club or business from both outside and inside interests that affect the cohesion of the team. If there is no contractual agreement or exchange of money for services, I don't see where the sensei is obligated to keep any person on a team whose activities outside the club have a negative impact on the operation of the club she created and manages.
And it's simply folly to believe that what one does in spare time will have absolutely no effect on other areas of life. What you do in your spare time can most definitely affect your career, for example. News articles abound warning that employers do google prospective employees to see how they behave online and there are stories aplenty of people losing jobs because of a lack of discretion in what they publish online. Pole dance at night and you'll probably lose your job as an elementary school teacher. Write pron on the side and you won't be welcome here at all no matter how polite you appear.
So, you're telling me that if I make my living making porn that I wouldn't be welcome here? That's none of your business and only illustrates why I have stopped posting. An internet forum is that. An INTERNET forum. It's not a job for members. It's not some jewel to be awarded only to those you find worthy. If you don't like that, you're in the wrong line of work. I have 2 webforums of my own, one is specifically for professionals. Do I go in and read their profiles and backgrounds to make sure none of them do something I don't agree with in my spare time? No. That would be insane and hugely inappropriate. I am HUGELY disappointed to see such a response from the owner of this site, much less someone who prides them self on being well versed in etiquette.
All of this has nothing to do with group loyalty. It's about a nosy leader who can't take anyone wanting to be friends with a group of people without having to go through the instructor to talk with those friends. It's about not wanting their personal messages between those friends read/over heard, and about not having to censor one's self because who knows what that instructor will decide is inappropriate from one day to the next. A group of friends, regardless of where they met and how can and do moderate themselves on their own. Presumably everyone involved are adults. How many threads have been here about how you should cut out toxic people in your life? How is having a group of friends where they come and go any different? Not to mention they're not affiliated with said group. Hell, there are more than likely people involved in that group that aren't even members of the original karate school.
Absolute power corrupts absolutely. That's not a "old saying" because it has no truth to it. I have left a group because of a leader and their "managers" not being able to be consistent and fair in their interactions with the group. In doing so, I lost track with a lot of the friends I made because they were solely from online interactions. In the day and age of facebook, you no longer have to worry about that. However, now people who are in charge of one activity are now trying to tell their MEMBERS (not employees) how they can spend their spare time and how interaction between other members should go? I don't think so.
I'd say that the environment the Sensei is breeding is toxic and I wouldn't take it. OP, if I were in your shoes, I'd leave and never look back. Along with all your buddies that decided they wanted to get together for what amounts to a cup of coffee online.