How to use radio buttons in gtkmm3?

Question

The documentation of GTK+ and gtkmm is bad in my opinion. All it says are things like "this function is deprecated".

Also, I can't find examples that show the exact way of how to use things the new way.

Can someone please show me a fully not-deprecated example using gtkmm3 code that features a popup menu with two different sets of radio buttons and a function that calls set_active(true) on one of the buttons?

Here is what I have so far:

m_refActionGroup = Gio::SimpleActionGroup::create();
insert_action_group("PlacepieceMenu", m_refActionGroup);
// These are part of one set of radio buttons:
m_refActionGroup->add_action("PlacepieceWhitePawn", sigc::mem_fun(*this, &ChessPositionWidget::on_menu_placepiece_white_pawn));
m_refActionGroup->add_action("PlacepieceBlackRook", sigc::mem_fun(*this, &ChessPositionWidget::on_menu_placepiece_black_rook));
// These should be part of another set of radio buttons in the same menu:
m_refActionGroup->add_action("ToMoveWhite", sigc::mem_fun(*this, &ChessPositionWidget::on_menu_to_move_white));
m_refActionGroup->add_action("ToMoveBlack", sigc::mem_fun(*this, &ChessPositionWidget::on_menu_to_move_black));

m_refBuilder = Gtk::Builder::create();
m_refBuilder->add_from_file("/home/carlo/projects/cwchessboard/PlacePiece.glade");

As far as I can tell, the glade file doesn't say which menu item belongs to which radio button group. In fact, it doesn't say anything about what the menu items are at all; here is a piece of the glade file:

    <child>
      <object class="GtkMenuItem" id="PlacePieceWhitePawn">
        <property name="visible">True</property>
        <property name="can_focus">False</property>
        <property name="action_name">PlacepieceMenu.PlacepieceWhitePawn</property>
        <property name="label" translatable="yes">White pawn</property>
        <property name="use_underline">True</property>
      </object>
    </child>

Also, no matter what I try, I can't call set_active on any of the items.