Kenjutsu Shoujo Izumi Rei to Mujintou ~Ouchi no Okite da... Omae to Kozukuri ni Hagende Yaru~

The protagonist is a school pupil, and the successor to the Yashima school of martial arts.

This year, he' is determined on getting a girlfriend to enjoy summer with!

But just as he has made up his mind, his father sends him to a remote uninhabited island to duel against a student of a rival martial arts school.
On the island, he meets Izumi Rei, a cool and beautiful girl. They duel, and Satoshi narrowly emerges victoriously.
The girl then tells him that since she lost, by the rules of her house, she must accept Satoshi's children.


[From English DLSite]