Josh McKoon, the chair of the Georgia Republican Party, boasted at the state convention in late May that he had “very good news.”

Georgia Republicans had just orchestrated a takeover of the state election board, an unelected body that sets voting rules. With this new majority, Republicans could enact an agenda that would help former President Donald J. Trump win in November, Mr. McKoon said.

“I believe when we look back on Nov. 5, 2024, we’re going to say getting to that 3-2 election-integrity-minded majority on the state election board made sure that we had the level playing field to win this election,” he said.

Since the takeover, the Georgia State Election Board has approved a host of rules on certifications and investigations backed by right-wing election activists who claim, falsely, that the 2020 election was stolen from Mr. Trump. The moves underscore a sharp rightward turn for what is supposed to be an apolitical body and have alarmed Democrats, election officials and even some Republicans.

“Clearly, the Trump allies have learned their lessons from the failure of the attempted coup of 2020, and they’re starting earlier and attempting to burrow more deeply into the most vulnerable pieces of the election system,” said Norm Eisen, a longtime Washington lawyer and chair of the State Democracy Defenders Fund, a nonpartisan election watchdog group.