The Ise clan (伊勢氏)

The Ise clan is a Japanese clan.
There were several lines as follows:

The Kanmu-Heishi (Taira clan) family
This is explained in detail in this article.

The clan of Kodama Party, one of the Musashi-shichito Parties (seven parties of samurai in Musashi Province)
This clan descended from the ancient Arimichi clan.

The Kanmu-Heishi (Taira clan) lineage of the Ise clan is thought to have descended from the TAIRA-no-Korehira family pedigree. The Ise clan took charge of civility and rules of etiquette of samurai families, and the clan taught the art of manners to the legitimate sons of the Ashikaga Shogunal family for generation after generation during the Muromachi bakufu period.

Relations with the Ashikaga clan dates back to the Sadatsugu ISE in the period of the Northern and Southern Courts (Japan). Sadachika ISE served as a tutor to the eighth Shogun Yoshimasa ASHIKAGA, which resulted in a significant influence over politics as well. Sadamune ISE, a child of Sadachika, and Sadamichi ISE, a grandson of Sadachika, cooperated with Masamoto HOSOKAWA who was from the rising Hosokawa clan and continued to have the influential position in the bakufu.. A grandchild of Sadamichi, Sadataka ISE and his son, Sadayoshi ISE, served the 12th Shogun Yoshiharu ASHIKAGA and the 13th Shogun Yoshiteru. During the period of Yoshiteru, they exclusively controlled the shogunate government with Nagayoshi MIYOSHI who increased in power after the decline of the Hosokawa clan, but later the Ise clan conflicted with Nagayoshi and lost their position, weakening their power. However, when Yoshiaki ASHIKAGA and Nobunaga ODA went to Kyoto together, Sadayoshi's children, Sadatame and Sadaoki served as bailiffs. Due to Sadatames' illness, he transferred the headship of the family to Sadaoki. Sadaoki didn't go to Bingo with Yoshiaki, but he became a yoriki (a police sergeant) for Mitsuhide AKECHI. After the Honnoji Incident, Sadaoki ISE participated in the Battle of Yamazaki as an Akechi force, and Sadaoki ISE attacked and conquered Nobutada ODA at Nijo-gosho Palace, but later, Sadaoki was eventually lost to the Hashiba force. A child of Sadaoki, Sadahira ISE served the Tokugawa clan during the Edo period, and their offspring continued to serve as a taishin hatamoto (greater vassal and a direct retainer of the Shogun) family.

The founder of Gohojo clan, a daimyo (Japanese territorial lord) in the Sengoku period, Sozui ISE (normally known as Soun HOJO), belonged to the Ise clan. Although there were several theories as to his roots, scholars believe he was from a branch family Ise clan of Bitchu. It is said that the family of Soun started to call themselves 'Hojo' after Soun's death, in the period of his son Ujitsuna HOJO.
Therefore, strictly speaking, it would be correct to call the Hojo family in the days of Soun's 'Ise family.'