A few months ago, Attorney General Jeff Sessions recused himself from the FBI's Russia investigation, so Deputy Attorney General Rod Rosenstein became the Acting Attorney General for purposes of the Russia investigation. And Rosenstein appointed a special prosecutor, Robert Mueller, to oversee the investigation. Now the Washington Post is reporting that the White House is considering either firing Attorney General Sessions, or asking him to resign, as "part of a strategy to fire special counsel Robert S. Mueller III and end his investigation into whether the Trump campaign coordinated with the Kremlin to influence the 2016 election".
My question is, what would happen if President Trump either fired Sessions or Sessions resigned? Who would oversee the Russia investigation? At least temporarily, I assume Deputy Attorney General Rosenstein would become Acting Attorney General, not just for the Russia investigation but for the entire Department of Justice. But what would happen once Trump appoints a new Attorney General to replace Sessions? Would the new Attorney General take over the Russia investigation from Rosenstein, or would Rosenstein continue to be Acting Attorney General for purposes of Russia investigation indefinitely?