When your Friendship Lamp is stuck on purple, it means it is updating.
But, if it seems to be updating for awful long time, you can do something to help it move forward.
If it is your normal home WiFi, please follow these steps:
Please turn your WiFi off at the router, and unplug and plug the lamp back in. At this point the lamp should turn blue and then dim red, meaning it is in WiFi setup mode.
In order to test that the lamp can connect, please connect the lamp to a mobile hotspot.
If the lamp connects to the mobile hotspot, please turn your WiFi on again (and turn your hotspot off) and try to connect the lamp to your WiFi.
You should now be able to connect to your WiFi, enter your GroupID and be connecting with your friends! :)