Is injecting a Factory into consumer a bad idea?

I have a factory to build cars... It a basic factory, it accepts name of a car, looks for all classes that implement ICar, chooses the correct type based on car name and initializes the car.

public class CarFactory
{
    Dictionary<string, Type> cars;

    public ICar CreateInstance(string carName)
    {
        // choose type of class from all classes that implement ICar 
        Type t = GetTypeToCreate(carName); 

        return Activator.CreateInstance(t) as ICar;
    }

    // some more code...
}

Now, I have a service which uses CarFactory. What is the right way to initialize CarFactory in my service?

Should I inject it? I feel this is the best solution except that Factory Pattern is a form of IoC itself and I am not sure if it is a good idea to inject a factory into the consumer class?

public class SomeService : ISomeService
{
    private CarFactory _carFactory;

    public SearchService(CarFactory carFactory)
    {
       _carFactory = carFactory;
    }

    // more code...
}

If injection of a factory is a bad idea, should I use static factory? or just instantiate it in the constructor?