For groups you’ll make sure that your server and your device supports groups.
Group support is device dependent! Some manufacturers don’t allow other groups than Google. They patched it in their proprietary Android system and disabled it. If you need group support you can however install Lineage OS - this is the real open source Android that still has contacts group support.
Also make sure that you set the correct contact group method for your server when setting up DAVdroid (in the last step you can set the group method). You can find information for all servers on our configurations page on you website.