Save Game really just means save data - you’d probably just need to save a single variable - int. You need to save which avatar they’re using because when you load a new level everything is wiped, so to speak.
Alternatively, you could store this information in the Game Instance (that persists) instead but if you restart the project, the selection will need to be done again. If you do not care about data persistence between sessions, the GI is probably enough.