TYPO3 Core

You could set this behaviour by Page TSconfig, e.g. like:
TCEMAIN.permissions.group = show,edit,delete,new,editcontent
TCEMAIN.permissions.groupid = 123

see: http://typo3.org/documentation/document-library/references/doc_core_tsconfig/4.1.0/view/1/3/

New group shouldn't be specified (as in example: 123), but transferred from the parent page.

Thus, this would be a new feature. Unfortunately it's not possible to just change this in the current code - but to integrate a new property for TSconfig, like e.g. the following:
TCEMAIN.permissions.groupid = parent

Right Oliver, and that should be default setting.

This is very important issue, should be implemented v4.2.

Here is the implementation for TCEMAIN.permissions.groupid = parent:

t3lib/class.t3lib_tcemain.php:730

FROM:

$fieldArray = $this->setTSconfigPermissions($fieldArray,$TSConfig['permissions.']);

TO:

$fieldArray = $this->setTSconfigPermissions($fieldArray,$TSConfig['permissions.'],$tscPID);

t3lib/class.t3lib_tcemain.php:5142

FROM:

function setTSconfigPermissions($fieldArray,$TSConfig_p)    {
        if (strcmp($TSConfig_p['userid'],''))    $fieldArray['perms_userid']=intval($TSConfig_p['userid']);
        if (strcmp($TSConfig_p['groupid'],''))    $fieldArray['perms_groupid']=intval($TSConfig_p['groupid']);

TO:

function setTSconfigPermissions($fieldArray,$TSConfig_p,$tscPID)    {
        if (strcmp($TSConfig_p['userid'],''))    $fieldArray['perms_userid']=intval($TSConfig_p['userid']);
        if (strcmp($TSConfig_p['groupid'],''))    {
            if($TSConfig_p['groupid']=='parent')    {
                $record=t3lib_BEfunc::getRecord('pages',$tscPID,'perms_groupid');
                $fieldArray['perms_groupid']=$record['perms_groupid'];
            } else {
                $fieldArray['perms_groupid']=intval($TSConfig_p['groupid']);
            }
        }

Instead of:

if($TSConfig_p['groupid']=='parent') {

it's better:

if(strcasecmp($TSConfig_p['groupid'],'parent')==0) {