Typing a context name into the Context column works for actions but not for action groups (actions with indented children beneath them).

You can, however, assign a Context to an action group through the Group Inspector. This is really a more powerful route, because you can select several action groups, and apply the same Context to all of them through the inspector.

From the menu Inspectors > Show Inspectors > Group

or start with the default Cmd-Shift-I shortcut.