I have not read Crime and Punishment and know very little about Dostoevsky (and I also never really looked into how much of the real person was infused into the character), but I just did a quick Wikipedia Adventure(tm) and I'm wondering if Crime and Punishment is a power that rules over life and death entirely. The power lets him just kill people as he pleases, which you may be able to correlate with the utilitarianism that Crime and Punishment was (apparently) written critical of, but he has also been shown to be possibly immortal and likely ageless, which means that his power likely isn't simply death. Additionally, Fyodor (the character) reflects parts of Dostoevsky (the author) in that he loves God greatly, which leads me to my next point.
Fyodor is himself providence incarnate and the first Ability User, given control of life and death in order to guide humanity. I can't think of any real evidence other than loving God, but it would be cool. The more reasonable thought I have is that Fyodor may be meant to mirror the Prodigal Son (which Dostoevsky apparently recounted to his children on his deathbed), having went out into the world and squandered a power that could help so many, instead committing sin and hurting others, although Fyodor is yet to return home (and this returning home part is what makes me think him having been given a grander purpose in life a more feasible possibility). His story is one of sin and forgiveness where Fyodor is doomed to an eternal life until humanity has been saved (under my crack theory) or until Fyodor "returns to his father," (which is what makes me want him to be the first and given reason from birth, as it implies a greater power which correlates to loving God's guidance) or both, likely dying afterwards. Going further into this Dostoevsky death parade, a possible self-sacrificial death meant to benefit others would connect to John 12:24, which is inscribed on Dostoevsky's grave, with Fyodor being alone in life (and if he's the first he'd have been a solitary existence), but sowing the seeds for the future in death.
To summarize: Fyodor's immortality is likely an aspect of Crime and Punishment, possibly expanding the power to full control over life and death rather than the ability to kill. Furthermore, in light of his immortal and ageless body, it is possible that he is one of the earliest Ability Users in history, and he has been wandering the world committing evil just for the hell of it ig. Additionally, when taken in tandem with what I've been led to believe Dostoevsky wrote about and his death, Fyodor may have later developments relating to his crimes that ends in self-sacrifice as a means of atonement and ending his immortal life.
This is very messy, I have a very poor memory of the finer details, and I have very little knowledge of how the story is written that may lead to the story being written like this, but food for thought nonetheless. Please insult me (and also actually correct me) if you think I'm just completely off base (for actual reasons)