The idea was to build a radio station powerful enough and near enough to the front, that the signal would reach German and European listeners. It began in 1942 under director Brewster Morgan from CBS and engineer Richard Condon. Station manager Saudek was from NBC, and the chief of the ever-important German-language desk Rober Nauer from from WLWO-AM. Work was halted that year as much of the initial team was pulled off ABSIE to build out AFRN. They didn't return to the job until the Fall of 1943. Despite the blitz, their London studios were ready by April of 1944.
It's hard to tell but when the Psychological Warfare Division (PWD) of SHAEF (Supreme Headquarters Allied Expeditionary Force) got involved things got more serious. SHAEF was founded in the spring of 1943 and their PWD department was keen on propaganda, radio propaganda and by extension ABSIE. SHAEF/PWD was headed by US Brigadier-General Robert A. McClure who had previously commanded the PWB (Psychological Warfare Branch) of AFHQ (Allied Forces Headquarters) for Eisenhower. It was SHAEF that oversaw airdropping remote transmitters into Zeesen, Obisfelde, Wiederau and Munich so all of Germany could receive the BBC and ABSIE.
RCA built ABSIE four medium wave transmitters and they leased six shortwave transmitters form the BBC. The 10 transmitters made them more difficult to jam and easier to receive on mainland Europe. Much of the staff came directly from OWI: French journalist Pierre Lazareff.. later the publisher of France-Soir, Alfred Puhan.. later a U.S. Ambassador and many others. These foreign language desks were core to it's mission. Reaching Europeans with vital information helped them organize resistance. At it's peak it had a multilingual staff of about 250 employees. It also rebroadcast popular BBC and VOA programs. More here.
Their last broadcast was on the 5th of July in 1945. Before they played Yankee Doodle, Director Elmer David made a statement declaring their mission accomplished and that the BBC and VOA would continue on. the book The Psychological Warfare Division, Supreme Headquarters Allied Expeditionary Force covers some of this in great detail.