How do I make it so that the activities I do on a certain group (or all the groups if the former isn't plausible) hidden, in that, it doesn't show up on my friends' news feed? I don't own or run the groups so I don't have control over their group types (Public, Private, Close).
You cannot do that. What your friend can and cannot see in his news feed is completely decided by your friend's settings, and you cannot do anything about it.
Speaking of the extreme, you can Block your friend on Facebook, then he/she would nnot be able to see any of your Groups and other activities.