Flutter MacOS : Additional implementation for sign in with google

I' ve developed a Flutter app based in Firebase Auth with the additional sign in method of Sign in With Google. I implemented it using the package google_sign_in: ^4.5.6 .

It works easily for iOS, Android and Web. For MacOS I used the same Certificate and added this code in the info.plist file:

<key>CFBundleURLTypes</key>
    <array>
        <dict>
            <key>CFBundleTypeRole</key>
            <string>Editor</string>
            <key>CFBundleURLSchemes</key>
            <array>
                <string>com.googleusercontent.apps.*****/Private key/*</string>
            </array>
        </dict>
    </array>

But it still doesn't work. The macOS app works perfectly using Firebase Auth and is also connected to the network. So:

1. Is there a more action to do or code to add (like WEB support)?
2. If I press the google sign in button nothing is shown it doesn't open the page with google log in and Also in the console it returns nothing.