Found, a Private Casket Carrier:

Air Mobility Command has identified Kalitta Charters of Michigan to fly military members’ remains from Dover AFB, Del., to their homes for burial, reports Stars and Stripes. AMC began seeking a private carrier following a complaint about handling of a casket by a commercial airline. AMC already has Kalitta under a $10 million contract for nonscheduled chartered passenger service. This new contract, according to the Stripes, is for up to $11 million and runs for six months from Jan. 1, while the Air Force works on a longer term contract.