Granting an extension
To grant an extension with anonymous marking turned on is a little less straightforward than non-anonymous marking however is certainly possible. As you can imagine, the Gradebook will not display user names so you will be unable to decipher which student is which and therefore, be unable to determine who requires an extension.
You will need to create a group containing students who require extensions (this can be done post-assignment creation if needs be).
Creating Groups and Groupings
Directions to do create groups and groupnigs can be found on the Groups and Groupings page
Allocating the Group to the Assignment
You should seek to create and populate a group within your Moodle course. This group should contain those individuals who require an extension and will enable you to differentiate them from the rest of the course participants (whilst still retaining anonymity).
To apply the group functionality to the Assignment activity, go to the activity settings, scroll to the section entitled Common module settings and select the Separate Groups within the Group Mode field. Once you have done this, remember to scroll to the bottom of the page and click on one of the Save buttons.
After applying setting, enabling extensions can be be granted to group members as appropriate by visiting the assignment submission overview screen, selecting the group name from the top left hand corner of this interface (as shown below).
Once the participants have been filtered in this way, members of the group can then be granted extensions as shown below.
Note: If students have been granted different extensions, you will require more than one group.
Clearly, true anonymity can only take place if more than one student is receiving an extension in this way.