Question

Which group believed the Islamic leadership should stay in Muhammad’s family?

Answer 1

The Shiites believed that Ali, Muhammad's son in law was the fourth rightly guided caliph, as he was related to Muhammad. This caused the first division in Islam as two groups, the Shiites and the Sunnis formed.

Answer 2

When Muhammad died, most of the Muslims chose Abu Bakr, a close Companion of Muhammad, as their political and social leader known as Caliph.  Abu Bakr was accepted as Caliph by a majority of the community because they saw the succession in a political way and not spiritual one. Another group, however, smaller and more self proclaimed correct, which also had in it some of the senior Companions, considered that the Prophet's son-in-law and cousin, Ali, should be Caliph. They understood that the Prophet had appointed him as the sole interpreter of his legacy, in both political and spiritual terms.

It was helpless though, because eventually, Abu Bakr was became First Caliph.

That smaller group was the Shiites.