stitch (something) onto (something else)
To fasten something onto something else with stitches. I stitched a patch onto my jacket to cover up the large the tear. I got my new jersey for the football team, but they haven't stitched my name onto it yet.
stitch something onto something
and stitch something onto sew something onto the surface of something else. Fred stitched the badge onto his jacket. Fred stitched on the badge.