The Saturn V[f] is a retired American super heavy-lift launch vehicle developed by NASA under the Apollo program for human exploration of the Moon.

NASA is not developing Relativity Space's rocket.

"On Tuesday, NASA said it hired the company to build a spacecraft to house a suite of scientific instruments, launch it into space, and fly it to Mars."

Plus, George Mueller, who managed the rocket team, worked for NASA, not some private company. So did all the engineers.

"The largest production model of the Saturn family of rockets, the Saturn V was designed at the Marshall Space Flight Center (MSFC) in Huntsville, Alabama. The program was managed by American George Mueller; technical design was led by scientists relocated from Nazi Germany, most notably Wernher von Braun, as well as Kurt Debus and Arthur Rudolph. This group had developed the first US launch vehicles, the Redstone rocket family, under the Army Ballistic Missile Agency. All engines were built by Rocketdyne. Boeing built the kerolox S-IC first stage powered by five F-1 engines; these remain the most powerful single chamber liquid-fuelled engines ever built. North American Aviation the hydrolox S-II second stage, and Douglas Aircraft Company the hydrolox S-IVB third stage, powered by five and one J-2 engines respectively. IBM and MSFC designed the rocket's instrument unit. "

https://en.wikipedia.org/wiki/George_Mueller_(engineer)