The Orpheus Society is a group that tried to learn more about the Problem. Though many people share this common goal, the Orpheus Society is kept quite confidential and is far ahead technology wise.
The society's name comes from Orpheus, a musician in Greek mythology, and its symbol was inspired by Orpheus's lyre. In the mythology, Orpheus traveled to the underworld in an effort to take back his dead wife. His music with his lyre was so beautiful that the lord of the underworld allowed his wife's soul to return. Thus, the society named itself from the man who successfully traveled in and out of the underworld/afterlife. And, it took its symbol from the lyre which Orpheus used to somehow defeat death.
The Orpheus Society aims to "subdue" ghosts and win against death as described by the Orpheus Society secretary. The secretary had not been lying but he did mean to sugarcoat their goals. The society's goal of "subduing" ghosts meant using the ghosts' ectoplasm in some way to their benefit like how Marissa kept her soul alive. And when the secretary said their other goal was to win against death, he meant them, the members, achieving immortality and never to be taken by death.
The group's advanced technology on ghosts were claimed, by the secretary, to have been collectively designed by the members. Although, Marissa and Ezekiel's input on the technology was implied. The society invented a great deal of weapons to fight ghosts like what the Rotwell Agency does. But unlike the Rotwell Agency, the Orpheus society kept the equipment to themselves. Their decision to be secretive about the equipment hints at their questionable motives.
Marissa Fittes, in the body of Penelope, was revealed to be the leader of the group. Marissa at one time personally lead the members in Dark London. The older than usual appearance of the Orpheus Society members may have been the result of their travels to the Other Side. The members must be aware of Ezekiel, who always accompanied Marissa, because of their travels in Dark London and of their crystal goggles. On Marissa being Penelope however, the members may not be aware.
Marissa's control over DEPRAC, some furnaces (the Clerkenwell Furnaces are owned by Fittes), illegal relics trade, and other agencies had allowed Marissa and the Orpheus Society to collect Sources of ghosts. Those Sources were used by Fittes and the Orpheus Society in opening gates to the Other Side. They had been doing the same process that when Donald and Celia Lockwood presented their findings about the Polynesian Shamans, who had been opening gates in a similar fashion the society does, the society feared for their secrecy. They arranged the murder of the couple who would later die in an "accident".