This is a visualization map of the second mission and per Roger's translation, "the green ball is Earth, the red one is Mars, and the large arrows show how the three fleets jointed up - the Jupiter one (lower left) and the one from the other colonies (upper left) join up with the Mars fleet, and they're also joined by more ships from other colonies in earth orbit (upper right). They attack Earth, and then the surviving flagships (giggle) head back to Mars."
So, Jupiter Base was part of the second attack wave and the red Zeta is not a custom. There is also the Second Mission Jupiter patch(also below) which would also indicate such and the fact that there are two "D-Day" invasion stripes on the wings of the Zeta. They just didnt show the Jupiter Base forces during the first episode for whatever reason
In the end, its just a plot element to give Fuke her own colored Legioss