Female Izuku in the world of High School DXD

I know that I had already made a story where Izuku is in the world of Highschool DXD, but this is my own twist on that version of events. Instead of Izuku being transformed into a Dragon, she will stay human until she is met with the members of the Gremory Household within the Underworld... where she will become a pure blood devil through a small ritual by Sirzechs Lucifer which takes half of his powers in order to complete. Izuku will become the descendant of Sirzechs Lucifer... albeit through a roundabout way to become a pure blood devil. For some reason... she really likes a certain Bael Clan member who is the strongest Young Devil.