Members of flagged revs 'reviewer' user group should be able to change the stable settings. Use case for this is to be able to stabilize pages without being in administrator user group. In example there could be bot which would dynamically stabilize pages with regular vandalism or temporarily stabilize pages with likely vandalism until they are reviewed. (eg. more or less same what is discussed in enwiki's RfC). Community discussion can be found here.
I think that the change is something like below and it is made to the flaggedrevs.php.
$wgGroupPermissions['reviewer']['stablesettings'] = true;