Github has an API to manipulate the release which is documented.

so imagine you want to get the latest release of ansible (which belong to the project ansible) you would

• call the url https://api.github.com/repos/ansible/ansible/releases/latest
• get an json structure like this

{
  "url": "https://api.github.com/repos/ansible/ansible/releases/5120666",
  "assets_url": "https://api.github.com/repos/ansible/ansible/releases/5120666/assets",
  "upload_url": "https://uploads.github.com/repos/ansible/ansible/releases/5120666/assets{?name,label}",
  "html_url": "https://github.com/ansible/ansible/releases/tag/v2.2.1.0-0.3.rc3",
  "id": 5120666,
  "node_id": "MDc6UmVsZWFzZTUxMjA2NjY=",
  "tag_name": "v2.2.1.0-0.3.rc3",
  "target_commitish": "devel",
  "name": "THESE ARE NOT OUR OFFICIAL RELEASES",
  ...
  },
  "prerelease": false,
  "created_at": "2017-01-09T16:49:01Z",
  "published_at": "2017-01-10T20:09:37Z",
  "assets": [

  ],
  "tarball_url": "https://api.github.com/repos/ansible/ansible/tarball/v2.2.1.0-0.3.rc3",
  "zipball_url": "https://api.github.com/repos/ansible/ansible/zipball/v2.2.1.0-0.3.rc3",
  "body": "For official tarballs go to https://releases.ansible.com\n"
}

• get the value of the key tarball_url
• download the value of the key retrieved just above

In ansible code that would do

- hosts: localhost                                                     
  tasks:                                                               

  - uri:                                                               
      url: https://api.github.com/repos/ansible/ansible/releases/latest
      return_content: true                                             
    register: json_reponse                                             

  - get_url:                                                           
      url: "{{ json_reponse.json.tarball_url }}"                       
      dest: ./ansible-latest.tar.gz       

I let you adapt the proper parameters to answer your question :)

The accepted answer solved the questioner's problem but didn't address the broader scope of the question.

How to install an Ansible module? The documentation is currently vague as to how to achieve this simple requirement!

An excellent general guide to writing modules (I've no connection to the author) can be found here.

The quickest way is to simply have a folder called library/ in the same folder as your playbook. Inside this folder, place the python script for the Ansible Module. You should now have a corresponding task available to your playbook.

If you want to share your module across multiple projects, then you can add an entry to /etc/ansible/ansible.cfg pointing to a shared library location, eg:

library        = /usr/share/ansible/library