which organisation was it a part of?
The group responsible was NASA's "Space Task Group".
Project Mercury
The group responsible for Project Mercury was NASA's Space Task Group, and the goals of the program were to orbit a crewed spacecraft around Earth, investigate the pilot's ability to function in space, and to recover both pilot and spacecraft safely. 
when did apollo 13 land on the moon It did not land on the Moon. It looped and returned safely to Earth what is this? Apollo 13 was the seventh crewed mission in the Apollo space program and the third meant to land on the Moon. what is the backdrop of this? In 1961, U.S. President, "John F. Kennedy" challenged his nation to land an astronaut on the Moon by the end of the decade, with a safe return to Earth. NASA worked towards this goal incrementally, sending astronauts into space during Project Mercury and Project Gemini, leading up to the Apollo program. what is the former project all about? Project Mercury was the first human spaceflight program of the United States, running from 1958 through 1963.
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Project Mercury [SEP] Organization
T. Keith Glennan had been appointed the first Administrator of NASA, with Hugh L. Dryden (last Director of NACA) as his Deputy, at the creation of the agency on October 1, 1958. Glennan would report to the president through the National Aeronautics and Space Council. The group responsible for Project Mercury was NASA's Space Task Group, and the goals of the program were to orbit a crewed spacecraft around Earth, investigate the pilot's ability to function in space, and to recover both pilot and spacecraft safely. Existing technology and off-the-shelf equipment would be used wherever practical, the simplest and most reliable approach to system design would be followed, and an existing launch vehicle would be employed, together with a progressive test program.