In The Fragrant Flower Blooms With Dignity Episode 4 titled “Warmth of the heart“, Subaru Hoshina asks Rintaro to stop meeting Kaoruko.
Back in episode 3, when she was introduced in the anime, she sounded cold and strict.
But the reason why she is like that when it comes to Kaoruko is explained in the latest episode. Subaru is actually trying to protect Kaoruko, and her concerns are legit.

Subaru Is Worried About Kaoruko’s Reputation
Subaru’s main reason is related to the school rivalry. Kikyo Academy and Chidori High do not get along. Kikyo students see Chidori students as reckless or undisciplined, and even the teachers there hate Chidori students.
If Kaoruko continues spending time with someone from Chidori, it could hurt her image.
Subaru knows how hard Kaoruko studied to get into Kikyo Academy. If even one rumor spreads, it could undo all of that. Subaru is trying to prevent this from happening before it becomes a problem.

Subaru’s Childhood Made Her Distrust Boys
The Fragrant Flower Blooms With Dignity Episode 4 also reveals Subaru’s past. When she was younger, some boys used to make fun of her just because of her silver hair, and that experience made her traumatized, afraid of boys, and unwilling to trust them easily.

She does know Rintaro is a kind person, but because Rintaro goes to Chidori, which is an all-boys school full of delinquents, Subaru finds it hard to believe he won’t cause issues for Kaoruko.
The major reason behind Subaru’s behaviour is, when Subaru used to get bullied by the boys, it was Kaoruko who stood by her side.

Since then, Subaru has always looked out for Kaoruko. She watched her study hard and finally get accepted into Kikyo as a scholarship student.
To Subaru, Kaoruko’s happiness and future are something worth protecting no matter what.
Even if there is only a small risk of Rintaro dragging Kaoruko into trouble or hurting her image, which could result in expulsion from Kikyo, Subaru wants to prevent it. That is why she asks him to stay away.
Rintaro Understands Her but Refuses
After listening to everything Subaru says, Rintaro does not argue. He understands why she feels that way. He even respects how much Subaru cares for Kaoruko.
But still, he politely tells her that he cannot stop seeing Kaoruko just because of that and apologizes to Subaru!