If you are not getting moderation messages:
Check your Backscatter spam settings. If it's set to discard, then the moderation messages aren't making it to you. The way that Groups.io sends messages needing moderator approval triggers FastMail's backscatter filter, which is separate from it's spam filter.
If you don't want to change the setting from discard, you can use a bit of custom Sieve code (under Rules in settings) to move the moderation messages before the backscatter discard rule kicks in.
Put something like this before the "3. Sieve generated for spam protection" section, being sure to adjust the
# Groups.io moderation messages get identified as backscatter by FastMail.
# This runs before the backscatter rule and stops processing, so they end up
# in the right folder.
address :is ["from"] ["firstname.lastname@example.org"]
, header :matches :comparator "i;ascii-casemap" ["Subject"] ["Message Approval Needed*"]
# Adjust this folder path as needed.