When my printshop lets their printer print an image with an embedded sRGB profile, and no CMYK based profile, with the printer managing the colour mapping itself it always does a good job. If instead they print using their chosen FOGRA27 CMYK profile, it's normally okay, but for some images it gets the blue/greens wrong.
But they will not do batch jobs as printer managed jobs because they insist the printer will not consistently output the same throughout the batch, whereas it will if they use the FOGRA27Coated profile.
But why would this be so?