Even better would be to separate the aiki core and the custom site into separate folders in the root web directory.
/srv/www/example.com/public/aiki « contains aiki core code
/srv/www/example.com/public/framework « contains empty directory structure for custom site
Your webserver would likely point to /srv/www/example.com/public/framework/index.php or something. The config file would be in framework/, which would specify the path to the aiki/ folder. Therefore, aiki could even be located at /usr/local/lib/aiki/. Your copy of stable aiki/ could be in its own git repo. Your custom site at framework/ could be stored in a separate repo.
This is how Lithium does things: http://lithify.me/docs/manual/getting-started/installation.wiki