Why a module's singleton method is not visible in downstream eigenclasses where it gets mixed?

I understand the regular method lookup path i.e. class, superclass/module, all the way up to BasicObject. I thought it was true for singleton version of the chain also but doesn't seem the case when you mixin a module in the meta-chain. I'd appreciate if someone can explain why in the following example Automobile module's banner method is called instead of its singleton version when I have included this module in Vehicle's eigenclass.

module Automobile
  def banner
    "I am a regular method of Automobile"
  end

  class << self
    def banner
      "I am a class method of Automobile"
    end
  end
end

class Vehicle 
  def banner
    "I am an instance method of Vehicle"
  end

  class << self
    include Automobile
    def banner
      puts "I am a class method of Vehicle"
      super
    end
  end
end

class Car < Vehicle
  def banner
    "I am an instance method of Car"
  end

  class << self
    def banner
      puts "I am a class method of Car"
      super
    end
  end
end

puts Car.banner

# I am a class method of Car
# I am a class method of Vehicle
# I am a regular method of Automobile

First of all, include does not include eigenclass methods as you might expect. Consider:

module Foo
  class << self
    def do_something
      puts "Foo's eigenclass method does something"
    end
  end
end

module Bar
  include Foo
end

puts Bar.do_something
# undefined method `do_something' for Bar:Module (NoMethodError)

Note that this is consistent with the behavior of classically defined class methods:

module Foo
  def self.do_something
    puts "Foo's class method does something"
  end
end

module Bar
  include Foo
end

puts Bar.do_something
# undefined method `do_something' for Bar:Module (NoMethodError)

A common idiom is to define the class methods in a submodule and then trigger a call to extend when the module is included:

module Foo
  def self.included(base)
    base.extend ClassMethods
  end

  module ClassMethods
    def do_something
      puts "Foo::ClassMethod's instance method does something"
    end
  end
end

module Bar
  include Foo
end

puts Bar.do_something
# Foo::ClassMethod's instance method does something

The second thing to note is, that you are really including the instance methods of Automobile into the eigenclass of Vehicle, thus the instance methods of Automobile turn into (eigen)class methods of Vehicle.

Your Car class basically has nothing to do with all this. The only thing to note here is, that class inheritance also makes class methods available, whereas include does not. Example:

class Foo
  def self.do_something
    puts "Foo's class method does something"
  end
end

class Bar < Foo
end

puts Bar.do_something
# "Foo's class method does something"