Groberg experienced much difficulty in getting to Tonga: he was prevented from arriving by strikes, visa problems, and transport issues. Groberg served briefly in Los Angeles, Samoa, and Fiji while waiting for his transport to be finalized. When he finally arrived in Tonga, his first assignment was on the remote island of Niuatoputapu, which had had only limited contact with the outside world in the form of an occasional telegraph and a visiting boat. During the year he spent on the island, Groberg suffered from mosquitoes, a typhoon, and starvation. His missionary companion on Niuatoputapu was Feki Po'uha, who would later serve as district president in Niue, while Groberg was president of the church's Tongan Mission (which at that point included Niue).
After a year on Niuatoputapu, Groberg was assigned to more developed islands and served as a district president supervising smaller branch congregations of the church. Groberg later reported that the branches he dealt with lacked unity and morality. He had little contact with his supervising mission president and nearly drowned when pushed out of a boat during a major storm; he also suffered from exhaustion frequently. Groberg was denied an extension to his mission that would have allowed him to accompany a group of church converts to the New Zealand Temple. He flew home and shortly after married Jean Sabin; together, they have had 11 children.
Groberg later returned to Tonga as the mission president and later as a regional representative. He wrote a book about his mission from his memoirs called, In the Eye of the Storm, which was adapted into the 2001 Disney film The Other Side of Heaven.
In April 1976, Groberg became a general authority of the LDS Church. In the mid-1990s, he was president of the church's Asia Area, where he was closely connected with the initial sending of church missionaries into Cambodia. He later served as president of the church's Utah South Area, where he was responsible for initiating programs for missionary work among the Latino population there, and attempts to ensure that English-speaking wardshome taught the Latino members within their boundaries, even if they attended separate Spanish-speaking congregations. Groberg also served as president of the North America West Area from 1990 to 1994. In May 1992, Groberg presided over the organization of the San Francisco California East Stake, the church's first Tongan-speaking stake in the United States.
1. Emeritus general authorities are individuals who have been released from active duties as general authorities. However, they remain general authorities of the church until their death. Except for the three identified former members of the Presiding Bishopric, all living emeritus general authorities are former members of the First Quorum of the Seventy.
2. Former members of the Presiding Bishopric.