After much difficulty in getting to Tonga with strikes, visa problems, and problems finding boats, he served briefly in Los Angeles, Samoa, and Fiji while waiting to get to Tonga. When he finally arrived in Tonga, his first assignment was on a remote isolated island called Niuatoputapu, which with neighboring Tafahi, was a place whose contact to the outside world was an occasional telegraph and visiting boat. On this island he suffered mosquitoes, a typhoon, and starvation. He spent nearly a year there. While on Niuatoputapu, Groberg's companion was Feki Po'uha, who would later serve as district president on Niue, while Groberg was president of the church's Tongan Mission, which at that point included Niue.
He then returned to more developed islands and served as a district president supervising smaller congregations called branches. The branches he dealt with lacked unity and morality. He had little contact with his mission president. He nearly drowned when pushed out of a boat during a major storm and suffered exhaustion many times. He was denied an extension so he could take a group of LDS converts to the Hamilton New Zealand Temple. He flew home and shortly after married Jean Sabin; together, they have had 11 children.
He later returned to the Tongan islands 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. His missionary experiences in Tonga became the subject of the Disney film The Other Side of Heaven.
In April 1976, he was called to be a general authority of the LDS Church. The changes he saw in Tonga from a missionary to church leader were documented in his book The Fire of Faith. He had other assignments besides Tonga as a general authority. In the mid-1990s he was president of the church's Asia Area, where he was closely connected with the initial sending of Mormon 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 make sure that English-speaking wardshome taught the Latino members within their boundaries, even if they attended Spanish-speaking congregations. He 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 first Tongan speaking stake of the church in the United States. The North America West Area encompassed California and Hawaii.
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.