This Google Script updates the Twitter user’s profile image or avatar. Twitter asynchronously processes the uploaded file before updating the user’s profile image URL.

function oAuthConfig() {
  var oAuthConfig = UrlFetchApp.addOAuthService('twitter');
  oAuthConfig.setAccessTokenUrl('http://api.twitter.com/oauth/access_token');
  oAuthConfig.setRequestTokenUrl('http://api.twitter.com/oauth/request_token');
  oAuthConfig.setAuthorizationUrl('http://api.twitter.com/oauth/authorize');

  oAuthConfig.setConsumerKey('PUT CONSUMER KEY HERE');
  oAuthConfig.setConsumerSecret('PUT CONSUMER SECRET HERE');
}

function setProfileImage() {
  oAuthConfig();

  // This is a picture that will be set as Twitter avatar
  var picture = UrlFetchApp.fetch('https://twitter.com/image.png');
  var encodedImage = Utilities.base64Encode(picture.getContent());

  var options = {
    method: 'post',
    oAuthServiceName: 'twitter',
    oAuthUseToken: 'always',
    payload: { image: encodedImage, skip_status: true },
  };

  var request = UrlFetchApp.fetch('http://api.twitter.com/1.1/account/update_profile_image.json', options);
}