The actions are logged to the moderator log ONLY if the user managing the group is a moderator. This is by Xenforo design. It only logs actions from moderators.
If you have assigned the group moderator role to regular users, then you will need an extra add-on to support writing to the moderator log: Log all moderator actions
This add-on has originally been developed by NixFifty for one of my forums with some tweaks by me. It has been tested on a standard XF 2.1 (2.1.2) installation with default theme and no add-ons. Although I will try to help you with any problems, it comes 'as is' without any warranties. Always test add-ons before installing them on your production site. I will consider updates and future improvements if needed. Do let me know in this topic.