group skills carry over, but make sure the group you're transferring to –– is at least the same (or more) in group skills level than your previous group.
also, group contributions do not transfer with you to your new group. only group skills.

your max group skill would be that of your new group.
i.e.
Old group max group skill was lvl 40
You maxed till lvl 40
You left old group.
You transferred to new group with max group skill lvl 35
Your new group skill lvl is 35 (+ 5 overflow, which can't be used, not until your new group upgrades to group skill).
But, if your new group's group skill upgrades, yours would automatically go back to 40 (without doing the whole waiting and spending contribution points).

I hope this was easy to understand :'(