President-elect Donald J. Trump has been known to cast a wide net when seeking advice for hiring decisions. As his team ramps up the presidential transition process, Mr. Trump is calling friends and associates for input on who should be part of his administration, which he says will radically reshape the federal government.

But a group of aides and advisers, some of whom worked in the first Trump administration and others who are newcomers, have particular influence as the president-elect starts choosing his cabinet and setting his administration’s agenda.

The group of course includes his vice president-elect, JD Vance. Less known are other influential business executives and Republican operatives advising Mr. Trump. Over the weekend, the group began to navigate the ideological differences of the party — and Mr. Trump’s impulses — as they began setting on appointments, including Representative Elise Stefanik, Republican of New York, as U.N. ambassador and Thomas D. Homan as border “czar.”

“President-elect Trump will begin making decisions on who will serve in his second administration soon,” Karoline Leavitt, a spokeswoman for his presidential transition, said on Sunday. “Those decisions will be announced when they are made.”

Here are some of the key people to watch as they bring their influence to bear on steering the next administration:

Elon Musk

After investing more than $100 million in Mr. Trump’s campaign, the billionaire has gained tremendous access to the president-elect. Mr. Musk was around Mar-a-Lago, Mr. Trump’s club and residence in Florida, last week as the president-elect began his first formal transition meetings.

On Wednesday, Mr. Trump handed the phone to Mr. Musk while speaking to President Volodymyr Zelensky of Ukraine. Mr. Trump has also said he wants Mr. Musk in a role focused on slashing government spending.