Story

Second Eruption
The Core of Fire first formed when the Will of the Honkai gave it to Sirin during the Second Eruption, it was and dubbed the Gem of Haste. Through Sirin the Core's Soul was supressed and the Authority was utilised many times. After Sirin's death her corpse and the Cores held by it were later found in the ruins by Schicksal.
Schicksal
While the Core was in the hands of Schicksal it was experimented on and implemented into the altered battlesuit Divine Key known as the Key of Blankness to become the prototype Godsbane battlesuit Vermilion Knight: Eclipse. This was intended to allow users to wield the powers of the Herrscher of Fire and was used as a design basis for the other battlesuits in the Godsbane line.
Battle of Schicksal
During the Battle of Schicksal, Himeko through following Fu Hua's instructions she located the Vermilion Knight: Eclipse battlesuit. Upon putting it on Himeko gained a connection with the Core, the Herrscher of Fire soul within attempted to imprint on Himeko but due its suppression, along with the battlesuit not providing a direct connection and some help from Fu Hua using Fenghuang Down, Himeko was able to gain control over the Core and began imprinting onto the Soul. With use of the battlesuit and Core Himeko was able to go head to head with the Herrscher of the Void. After a fierce battle against the Herrscher of the Void, Himeko nullified most of the Honkai Energy in Kiana's body let her take it back, but at the cost of Himeko's own life. Due to being connected through the battlesuit the imprinting process took longer than usual and did not complete resulting in her not fully imprinting on the Soul before she passed, leaving a partial version of Himeko in the Core as the Soul.
Herrscher of Dominance
After floating around in Imaginary Space for a while the Core of Fire was found by the Herrschers of Dominance leading them to collect it and attempt to gain the Authority from it, however due to the nature of their Authority the Core of Fire needed a host so they planned to use it as a psychological weapon against Kiana to allow the Herrscher of the Void to reawaken and wield the Core. After the Herrschers of Dominance captured Kiana they showed the Core of Fire to her which broke her mentally for a small period of time as it meant that Himeko was truly dead. Kiana was able to obtain the Core from them fairly easily but due to the opening given the Herrscher of the Void reawoken and took in the Core of Fire to use the Authority from and once again supress the Soul. After a mental struggle with the Herrscher of the Void, Kiana drew upon the Authorities of Void and Fire to help her defeat the Herrscher and become the True Herrscher of the Void. After Kiana became the True Herrscher of the Void she connected with the Soul in the Core and got to have a final conversation and goodbye to Himeko.
Herrscher of Flamescion
Once Kiana became the True Herrscher of the Void she channelled the Authorities of Death and Fire from their Cores actively and became known as the Herrscher of Flamescion. Due to the continued use of the Core of Fire by another Herrscher, the Soul within continued to be further supressed. The Soul within the Core supported Kiana's decision though and supported her in the fight against the Herrscher of Dominance.
Herrscher of Finality
Once Kiana made the choice to become an artificial Herrscher of Finality she drew the Authorities from all the Cores with help of the Herrscher of Origin, through this she was able to get close enough to the Herrscher of Finality that the Cocoon of Finality granted her access to the Authority of Finality. Due to being artificial Kiana continued to require use of all other Authorities to wield the Authority of Finality, so the Herrscher of Fire Soul continued to be supressed within the Core.