Add Google Authentication to an Umbraco 9 website

GoogleAuthenticationExtensions.cs

using Umbraco.Cms.Core.DependencyInjection;
using Umbraco.Extensions;
using Umbraco.Cms.Web.BackOffice.Security;
using Microsoft.Extensions.DependencyInjection;
using Microsoft.Extensions.Configuration;

namespace Umbraco.Cms.Web.UI.NetCore.Configuration
{
    public static class GoogleAuthenticationExtensions
    {
        public static IUmbracoBuilder AddGoogleAuthentication(this IUmbracoBuilder builder)
        {
            builder.AddBackOfficeExternalLogins(loginOptions =>
                loginOptions.AddBackOfficeLogin(new BackOfficeExternalLoginProviderOptions(
                    "btn-success",
                    "fa-google"
                ), authOptions => authOptions.AddGoogle("Umbraco.Google","Google", options =>
                {
                    IConfigurationSection googleAuthNSection = builder.Config.GetSection("Authentication:Google");
                    options.ClientId = googleAuthNSection["ClientId"];
                    options.ClientSecret = googleAuthNSection["ClientSecret"];
                })));
            return builder;
        }
    }
}

Startup.cs

// Add it to your startup like so
public void ConfigureServices(IServiceCollection services)
{
    services.AddUmbraco(_env, _config)
        .AddBackOffice()
        .AddWebsite()
        .AddComposers()
        .AddGoogleAuthentication()
        .Build();

}

And for members? Not backoffice.

Sadly Umbraco doesn't support external login for members yet. I've raised a feature request for it, add your support if this is something you need. umbraco/Umbraco-CMS#10952

The discussion provides some hints on how to accomplish this. You will have to override MemberSignInManager and MemberManager.