LAMP lovers, HELP! Apache2 www-data permissions and PHP "permission denied" error

LAMP lovers, HELP! I'm confused. This works fine with my apache2/PHP config:

% pwd
% /srv/www
 % chown www-data:webmins index.php
 % chmod u=r,g=,o= index.php
 % ls -l | grep index*
 -r--------  1 www-data webmins   19 2012-05-24 20:05 index.php  // runs on localhost

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BUT, if I add the user www-data (apache2 user, I think?) to the group webmins and change permissions to allow group read, like this:

% chown root:webmins index.php
 % chmod 040 index.php
 % usermod -aG webmins www-data
 % groups www-data
 www-data : www-data webmins
 % ls -l | grep index*
 ----r-----  1 root     webmins   19 2012-05-24 20:05 index.php  // fails

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The whole thing fails and I get a PHP "failed to open stream: Permission denied" error. What am I missing? I thought that by adding the www-data user to the group webmins, and then giving webmins read access would be essentially equivalent to the previous config. Where did I go wrong? I must just have a misunderstanding. Please help!

PS: It might be worth noting that I have the SGID bit set on /srv/www so that the group of any folder created under it is by default set to webmins.

% ls -l /srv | grep www
drwsrwsr-x 5 www-data subversion 4096 2012-05-20 08:03 svn
drwxrws--- 4 www-data webmins    4096 2012-05-25 12:34 www

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