Change Emoji Appearance

Currently Wordpress displays emoji unicode symbols using an icon set like this one http://emojione.com/#gallery but to be honest I don't like how they look so I wonder if it's possible to use another icon pack that looks more native, like the one on this picture: https://en.support.wordpress.com/emoji/#emoji

Emoji icons URL now looks like https://s.w.org/images/core/emoji/2/svg/1f61a.svg