How inject service in AutoMapper profile class

To solve your problem you just need to wire IUserManager in DI, and make sure UserProfile dependency is resolved.

public void ConfigureServices(IServiceCollection services)
{
    // ...
    services.AddSingleton<IUserManager, UserManager>();
    services.AddSingleton(provider => new MapperConfiguration(cfg =>
    {
        cfg.AddProfile(new UserProfile(provider.GetService<IUserManager>()));
    }).CreateMapper());
}

And having that said, I would probably try to keep single responsibility per class, and not have any services injected into mapping profiles. You can populate your objects just before the mapping instead. This way it might be easier to unit test as well.

It's better to use custom IValueResolver for this purposes because it is fully supports IServiceCollection integration (using AutoMapper.Extensions.Microsoft.DependencyInjection).

You may need to implement a custom value resolver:

public class UserViewModelValueResolver: IValueResolver<...>
{
    public readonly IUserManager userManager;
    public UserViewModelValueResolver(IUserManager userManager)
    {
        this.userManager = userManager;
    }
    ...
}

And the registration in services may be reduced to:

public void ConfigureServices(IServiceCollection services)
{
    // ...
    services.AddSingleton<IUserManager, UserManager>();
}

Then you can get a mapper instance inside a controller by injecting IMapper via a constructor.

Based on: AutoMapper: Handling Profile Dependencies using Custom Value Resolvers - Tech Net