I wouldn't have had to bring this up, but thanks to a negative review, here I am.
Yes, the skull modifier, especially the URL & Base64 value, is not working on 1.20.2 because Mojang decided to put some restrictions on the API.
As the nature of BetterGUI, the main plugin contains a few basic features, so it doesn't receive frequent updates. The addons, though, are updated frequently through the Addon List.
One of the addons that fixes the exception is XCross, which uses XSeries. You can use this addon to make the skull work on 1.20.2.
About the recent exception, it was this exception:
Code (Text):
Caused by: java.lang.NullPointerException: Cannot invoke "org.bukkit.profile.PlayerProfile.getTextures()" because "ownerProfile" is null
at org.bukkit.craftbukkit.v1_20_R2.inventory.CraftMetaSkull.applyToItem(CraftMetaSkull.java:148) ~[paper-1.20.2.jar:git-Paper-249]
at org.bukkit.craftbukkit.v1_20_R2.inventory.CraftItemStack.setItemMeta(CraftItemStack.java:659) ~[paper-1.20.2.jar:git-Paper-249]
at org.bukkit.craftbukkit.v1_20_R2.inventory.CraftItemStack.asNMSCopy(CraftItemStack.java:60) ~[paper-1.20.2.jar:git-Paper-249]
at org.bukkit.craftbukkit.v1_20_R2.inventory.CraftInventory.setItem(CraftInventory.java:99) ~[paper-1.20.2.jar:git-Paper-249]
The exception was a bug from Spigot & Paper (out of my control), which is fixed in recent builds.
Therefore, just update your server jar and it'd be gone.
You should also update your server jar regularly to receive bug fixes, or use a server auto-updater. You can check out MCServerUpdater.