TLDR Imamat History: Aga Khans
Imam Shah Hasan Ali
After the death of Imam Khalilullahi Ali, the Shah of Iran gifted Shah Hasan Ali with the provinces of Qom and Mahallat (in addition to his father's estate in Kahak), legal recognition of the Ismaili community within the Qajar Dynasty, the title of Aga Khan (meaning Brother-King), and one of his forty-six daughters to marry. Fath Ali Shah appointed his grandson, Muhammad Shah, as his successor, but his son rebelled after his death. The Imam lent his military expertise to Muhammad Shah during the ensuing civil war and was appointed Commander in Chief after he had won. However, the Prime Minister repeatedly insulted and discredited Hasan Ali in front of the Shah, eventually causing the Imam to retaliate. The Prime Minister took this oportunity to arrest Hasan Ali, but he was pardoned by Muhammad Shah. The Prime Minister continued his bullying, so the Imam ultimately left Iran, joining the Ismaili community in Sindh, where he joined the British government. Originally, Shah Hasan Ali wanted to reclaim his lost territory in Iran, but the Qajari forces were too strong. Instead, he moved to Bombay and then Calcutta to escape Persian influence. He settled back in Bombay after the death of Muhammad Shah, and worked to raise the Ismaili community in India (and breed horses). There was a group within the Ismaili community that didn't accept the Imamat of Shah Hasan Ali. They argued that Pir Sadruddin had converted the Khojas to Sunni Ismailism (which isn't a thing), not Shia Ismailism. This issue was brought before the British courts in Bombay and as a result, the Aga Khan was recognized as the legal leader of the Shia Imami Ismaili Community.
Imam Shah Ali Shah
Shah Ali Shah was born and raised in Persia, his father had hired mullahs from across Persia and Arabia to teach him. He married an Iraqi tribal princess and had two sons. After he moved to join his father in Bombay, his wife died and he remarried. His second wife also died and he remarried again, this time to the granddaughter of Fath Ali Shah and lived with her in Baghdad and then Karachi where they had his third son. He served as the President of the National Muhammadan Association and a member of the Bombay Legislative Council. After he became Imam (while still presiding over the National Muhammadan Association), Shah Ali Shah's first two children passed away in their 30s. After nine months of grieving, Shah Ali Shah died, leaving his eight year old son to be the next Imam.
Imam Sultan Muhammad Shah
Sultan Muhammad Shah became the Imam when he was eight years old. He was raised mostly by his mother, with some assistance by his uncle (whose daughter he later married). He was fascinated with European culture and often travelled to Europe for long periods of time, leaving the Ismaili community in India under the care of his mother, who handled personal and community issues, as well as administration within the congregation. Once, while the Imam was overseas, a group of Ismailis in Bombay seceded from the congregation, building their own mosque and burial ground and declaring themselves "Ithna Asheri Khojas" to the violent reaction of Ismailis. However, Sultan Muhammad Shah chastised his own followers (something he would continue to do) for this violence. He helped to found (and became the president of) the All-India Muslim League, which fought for a separate Muslim State, and represented India to the League of Nations. He presided over an Ismaili community in India (and an Indian Ismaili community in East Africa) and set about reinstitutionalizing Ismaili leadership and putting funds toward education for the community. Before his death, after 72 years of Imamat, Sultan Muhammad Shah appointed his grandson as the next Imam instead of his son, Aly Solomon Khan, an unprecidented declaration in this history. However, unlike with the Qajari civil war, Prince Aly Khan graciously accepted his son as the Imam.
Shah Karim alHusayni
Shah Karim alHusayni was studying in Harvard University when he was appointed as Imam. Abandoning his studies, Shah Karim alHusayni quickly focused his attention on the wellbeing of his community, paying particular attention to those in East Africa, as a wave of Africanization swept through the area. Anti-Arab revolutions in Zanzibar threatened previously thriving Ismaili communities. Dictator Idi Amin violently expelled non-Africans from Uganda, including a large Ismaili population. Shah Karim provided his own funds and worked with the Government of Canada to get many Ismailis out of these troubled areas. Shah Karim alHusayni also built interdenominational institutions to promote healthcare and education in developing communities, even those that don't have Ismaili populations, encouraging his own community to do the same. To accomodate a global community, Shah Karim alHusayni's head of operations was in France, but he moved it to Portugal after 80 years of being Imam to secure alliances.