This behavior is seen if you have a device with a shared line that has an extension mobility (EM) profile associated with that device. After you add a call pickup group, it does not work unless the call pickup group is added first and then the rest of the settings are done.
Use one of these two workarounds to resolve this issue:
Remove the extension mobility and shared line from the phone, add the call pickup group and then add back the shared line and extension mobility.
Add the call pickup group before you add any other settings to the phone.