Details: After defeating all eight trainers apart of the Eevee Quest, return to Kagetora in Royal Avenue. Once there, he will challenge you to a Battle to finish your quest, with none other than his Eevee.
Eevee User Kagetora - Level 57 Eevee
If you defeat Kagetora, he will give you the Eevium Z-Crystal. This Z-Crystal allows Eevee to use the Exclusive Z-Move of Extreme Evoboost instead of Last Resort, which increases all of Eevee's Stats by two stages.