Im not sure if youre still interested in a group study but here is a game I found that might work well
Game description (goal and rules): Shiritori is a game similar to “Word chain”. Players are required to come up with nouns beginning with the final syllable of the previous word. When a player cannot come up with a word, they lose.
Recommended age: People of all ages can play this simple game, but older teenagers and adults are likely to enjoy it more as they have a more extensive vocabulary than children.
Recommended number of players: Two or more players are needed to play Shiritori.
Average game duration: This game can last anywhere from fifteen or twenty minutes to an hour.
Grammar you can practice while playing: While the game is played using only nouns, you can use them in their nominative form to practice grammar.
Vocabulary you can practice while playing: You are guaranteed to learn some new words while playing this game! Here is an example of a word sequence that could come up in the game:
I have to admit, but I don’t personally find shiritori useful for learning, although it is fun. It’s usefulness is akin to wordle I think. If you want to play more regularly, there’s a thread on the forum here iirc or on renshuu.org there’s 2-3 games going 24/7 (beginner, kana, and kanji versions I believe).