Try the 'MultiClipper' custom transformer.This transformer is located on the hub and clips data or imagery with attributes to the required area of interest where the clipping areas overlap.When you add the writer, you can even fanout the output by 'fme_basename'.This will put the output clipped images into separate folders.

Note:You can add more that one image into the 'ClippeeInput' port.

You could even use two 'Multiclipper' transformers to run them separately.

The group by should work but you have to make an attribute on both the raster and the polygons that has the same name.Easiest way to do it is to create an attribute on the raster that has the same name as on the polygons and fill it with the fme_basename attribute value, then group on that in the Clipper.