Thanks to Kessler over at Invision for suggesting the correct syntax to use.
If you want to use different team icons for each skin you need to first:
make a folder called team_icons under public/style_images/nameofskin.
Second: Put your images in that folder.
Third: Go to your ACP> Members> Manage Member Groups> Edit Group
Fourth: For the Group Icon Image put
Fifth: Do that for each group you have an image for.