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In December 1961, the selection of female cosmonaut trainees was authorised by the Soviet government, with the specific intention of ensuring the first woman in space was a Soviet citizen. In February 1962, Kuznetsova was selected as a member of a group of five female cosmonauts to be trained for a solo spaceflight in a Vostok spacecraft. Although she worked as a secretary, she parachuted as a hobby, which she had taken up in 1958, and by 1961 was a regional and national champion. Even though she was only 20 years old, her skill and courage as a parachutist led the cosmonaut recruiters to select her for their programme. In the early days of her training, Kuznetsova was the favourite to become the first woman in space, but by late summer 1962 some failures in the physically and emotionally tough preparation regime resulted in her being removed from training.
The honor of being the first woman in space was eventually given to Valentina Tereshkova who was launched into Earth orbit in June 1963 aboard Vostok 6. Tereshkova's backup was Irina Solovyova, with Valentina Ponomaryova in a supporting 'second backup' role.
Despite her earlier difficulties, Kuznetsova was recalled to cosmonaut training in January 1965 to prepare for a spaceflight on a two-woman Voskhod 5 mission, but the project was cancelled before she had a chance to fly.
Kuznetsova finally retired from the cosmonaut program in 1969 when it became clear that no women would be included on future Soviet flights.