22 months is quite young for a playgroup, and if there isn't a set, structured routine, which children of a young age need, it may just be overwhelming for her. That's why preschool and daycare have a routine the children can rely on and find security in. If you can I would stay and watch (where she can't see you) to see what is going on, how she's interacting with the other children, they with her, and so on. Possibly there's a problem with sharing, getting tired or frustrated, etc. I realize that she may not be verbal enough to explain, but have you tried asking her why she doesn't want to go? If she's saying "no" to this, ask yourself is she saying "no" to everything or just this? She may have a real reason she's not liking it.
But remember, it has been only a month. My daughter was older, about 3 1/2, and screamed for me every day (she went 4 days a week) when I left her at preschool/daycare for the entire first semester I returned to college. Her teacher told me she would cry until I was out of sight, (she could see me through a chain link fence as they were having outside play when I dropped her off) then would settle down. It still rattled me in my first class, though, and I would often call to check on her but always found out she was fine. After the winter break when we returned about a month later she never cried for me again! Apparently she just wasn't quite ready to be away for me when we began but matured.
Dolls and stuffed animals, balls, Duplo or Mega Blox, small cars and trucks, little people, play food (Leap Frog Cook & Play Potsy is good) puzzles, something to ride on, push or pull, books, interactive toy laptops like the ones from VTech, (yes, we live in a technological world) etc., are all fun and educational in their own way. You want to offer a variety of things that allow her to explore, learn and get exercise. My almost 26 month old has his own soccer ball to kick and run after, loves his books, laptop, and ABC Text & Learn because it looks like a cell phone to him, and it helps him count and learn his ABC's, which he loves.
Hope this helps, and hang in there : )