The user will need to be a Content Editor in order to edit the Content within the Community Group, to make them a Content Editor navigate to:
  1. Core > Security > Content Editor Access 
  2. Select the Communities Tab 
  3. Click Add Content Access Group
  4. Enter a Group Name
  5. Within Find Users, search for the User(s) you would like to give Content Editor Access to
  6. Click the link Add to Group >> 
  7. Select the Year in the Dropdown and click Change year if it was not the one pre selected
  8. Select the Category
  9. Once the Category is selected you can select the Groups you want this Group to have access to
  10. To select multiple items, you may use the Ctrl-click and Shift-click functions.
  11. Click Add Access
  12. Click Save & Exit