Annihilatus

is the secret branch of the Russian Orthodox Church that deals with magic, specifically the disposal of spirits and such that shouldn't exist. To them, a wandering spirit should be eliminated, as it's 'something that shouldn't exist'. From volume 4, in Tsuchimikado's words, they are 'Ghost hunters' as opposed to the Necessarius being 'Witch Hunters'. Annihilatus normally forcibly erases demons, ghosts and such rather than trying to 'send them off' to the next world in a peaceful manner. Information about this organization is limited, with the only known members being Sasha Kruezhev and Vassilisa.

Unlike Necessarius, whose leader Laura Stuart is also the Archbishop of Anglican Church and therefore contributes to its role in its Church's structure, it's implied in Volumes 13.5 & 18 that Annihilatus is only a side group within the Russian Orthodox Church and does not have much influence over the Orthodox Church at all.