I want to be able to put pictures, arrows, lines, text boxes, etc in a
chart so if I were to copy or move the chart, all of the things I put
in the chart will go with it.
Forget grouping. Copy the object, select the chart, and paste. Or select the
chart, then click on the button on the Drawing menu to draw a shape. Or
select the chart, then go to the Insert menu and insert the picture. Each of
these techniques embeds the item within the chart.