If you create a full group via the friend invite (so it says "My group" at the top of the group list), you always go first if you join a battle together, but you can't assign sigils.
If you create a full group via the adventure party invite (so it says "Adventure party" at the top of the group list), you can pre-assign sigils, but you don't always go first.
Please make adventure party groups *also* go first, this seems obviously how it should work. The sigil assignment was a nice QOL update, should also have first turn bonus.