Select the Resources tool from the Tool Menu of your site.

To grant uploading permission to group members, to the right of the group folder, from the Actions drop-down menu, select Edit Folder Permissions.

This displays the folder permissions dialog box.

Note: You will need to make the folder a group folder in order to limit uploading permissions to a single group.

In the student column, select Create resources, Edit own resources and Delete own resources, and Access/create group resources, then click Save.

This allows the students that are members of the group to upload and edit content in the group folder.

Students that are members of the group will have an "Add" section available in the folder's Actions dropdown menu.