Kasrkin and Schola Progenium?

It is stated on all media I find that Kasrkin are classified as a variant of the Tempestus Scions known as Grenadiers. If this is the case, could a graduate from the Schola Progenium, if sent to the Scions, become a Kasrkin? Or just become one in general?

Would this make sense in the lore? Wanted to asked people who may be more knowledgeable.