The leadership of Annihilatus is separate from the Russian Orthodox Church unlike the Anglican Church, whose head, Laura Stuart, also leads Necessarius and therefore contributes to its role in its Church's structure. It is implied in Volumes SS1 and 18 that Annihilatus is only a side group within the Russian Orthodox Church and does not have much influence over it at all.
To Annihilatus, a wandering spirit should be eliminated, as it is "something that should not exist" in this world, as the soul is meant to go to heaven, hell or the purgatory, and not remain in the world. Annihilatus normally forcibly erases demons, ghosts and other such phenomena rather than trying to "send them off" to the next world in a peaceful manner, regardless of their circumstances. To do this, they won't hesitate to use Magic banned since Rasputin's corruption of the Russian government, and it isn't rare for them to destroy everything in the area their targets appear, leaving no remains and making major changes to the surrounding terrain. Though they carry weapons and interrogation tools that are powerful against malign spirits, Annihilatus is not suited for interrogating and hunting down humans, if ever the situation calls for it.