The Utah Department of Corrections manages convicted felons in Utah.
Edmunds is one of three finalists for the spot, left vacant by Tom Patterson.
If Edmunds is selected and accepts the position, the Summit County Council will declare a vacancy and give notice to the Republican Party liaison, according to Summit County Deputy Civil Attorney Dave Thomas.
"Then the liaison would have 30 days to bring a name to the County Council," Thomas said.
The Summit County Republican Central Committee, which consists of Summit County Republican Party Chairman Henry Glasheen, Glasheen's executive committee and the county Republican precinct chairs, would hold a meeting to choose a name, which would then be provided to the County Council
"I will call a meeting at the time the vacancy takes place, if it takes place," Glasheen said.
Glasheen added that he already has a replacement in mind, but did not disclose the name.
"We want to make sure that whoever is appointed is qualified," he said. "He has almost two years left, so that person would be acting as sheriff. We want very much for that person to be qualified."
Glasheen said the candidate will need to have qualifications to work with Summit County and the County Attorney's Office.
"We've had a really crappy record as far as our previous sheriff," he said. "Edmunds came in and cleaned up the Sheriff's Department pretty meaningfully. We want to make sure that progress continues. It might be the only time in the four years since I've been chair that the County Council and the Republican Party in Summit County will be on the same page."
After the committee brings a name to the County Council, the Council has five days to approve or reject the name.
"If they object to the name, we can provide them with another name, or we can have the governor make a decision," Glasheen said.
Gov. Gary Herbert interviewed the candidates for the state corrections post Friday. According to Ally Isom, government affairs director for Herbert, Herbert will take the time necessary to make a decision on who to appoint, and the name will then be forwarded to the Utah Senate for confirmation.
The other two candidates interviewed are Mike Haddon, the current acting director of the Department of Corrections, and Rollin Cook, who is retired after serving more than two decades with the Salt Lake County Sheriff's Office.