I don't know what "nest" means! I just wanted to move several actions into a folder I was going to create for specific action types, i.e. a folder for "shadow" making actions, a folder for HDR type actions, etc.
"Can I move multiple existing action sets into a new folder in photoshop 2022?"
"I just wanted to move several actions into a folder I was going to create for specific action types, i.e., a folder for "shadow" making actions, a folder for HDR type actions, etc."
Your first question asks about moving action sets (aka folders) into new folders. You can't.
Your second question asks about moving actions into folders (aka action sets). You can.
When you click the folder at the bottom of the panel, you create an actions set. When you create an action, PS asks which action set you want to put it it. In addition, you can drag and drop actions from one set to another (Option + Drag will copy.)
If thats what you mean, then yes, click the folder icon to create an action set > call it "shadow" > drag and drop actions into it (easiest if the actions are collapsed). Repeat for "HDR", et cetera.
Be sure to save your action sets (folders). Select the set (not an action) and use the panel menu to save the set. Then you can use the same menu to Load them back in if necessary.