uncoveted, undesirable, unwished for, unattractive:
He was given the unenviable task of informing the victim's family.
Advanced English Dictionary
+adjective [usually before noun] difficult or unpleasant; that you would not want to have: She was given the unenviable task of informing the losers. Antonym: ENVIABLE
Random quote: It is not fair to ask of others what you are unwilling to do yourself.: Eleanor Roosevelt