The account usually gets automatically connected to Google Play/Game Center when you first enter the game, if you gave your consent to this. You can check if your account is connected in the game settings. If you see the button “Disconnect”:
it means, that your game account is connected and your progress is automatically saved. And if your settings window looks like this:
then we advise you to tap the button “Connect”.
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Last Day on Earth: Survival Game GUIDE