The RSF was founded in 1959 by Major James Lethbridge-Stewart and Captain Marianne Kyle. (PROSE: The Schizoid Earth) Shortly after its establishment they battled the Bannermen in Wales, from whom they obtained artificial gravity and fast drive technology which allowed them to construct a base in space. (PROSE: The Face of the Enemy)
The RSF's encounter with the Great Intelligence brought them into contact with the Master's counterpart Koschei, who helped them battle the Intelligence at the expense of his TARDIS, which was ruined though the console was salvaged. The RSF tortured Koschei and Ailla, eventually killing her though it's unknown if she somehow survived through regeneration.
When the RSF reviewed recordings of the Doctor's TARDIS arriving in their universe, they realised they could do the same with Koschei's own TARDIS. They tortured him, forcing him to go through his lives rapidly until he had none left.
However, the RSF could not avoid being drawn under the control of James' power hungry father, Gordon Lethbridge-Stewart. By the end of the 1960s, Gordon had taken the reigns of power and alienated James through his ultranationalist policies. James fled to another Earth using technology stolen from the Axons. (PROSE: Night of the Intelligence)