They did; from a similar question a little while back:
The Allies, like the Axis, employed both overt and covert radio propaganda. Overt, or 'white', propaganda made no secret of its origin; broadcasters from Germany collectively nicknamed 'Lord Haw-Haw', primarily William Joyce, opened with the announcement "Germany calling!" leaving little doubt as to the source of the broadcast. On the Allied side the BBC operated across the world, with a European Service broadcasting in several languages using powerful transmitters to reach occupied Europe. While obviously aimed at bolstering the Allied cause BBC broadcasts generally aimed for accuracy and reliability; Douglas Ritchie of the European Service said: "If by propaganda is meant the deliberate spreading of false rumours and false information then I am confident that such a charge against the BBC cannot be sustained. If by propaganda is meant the arguing of a case then it is true that a part, but only a part - of our broadcasts to Europe were propaganda."
Covert ('black') propaganda on the other hand masqueraded as something else; for radio stations this typically meant broadcasts purporting to be from within the countries they targeted, using plausible frequencies, often in the range of official broadcasts. Germany set up stations including Radio Caledonia, aimed at Scottish listeners, and Workers' Challenge, a socialist station, spreading defeatist rumours of decreasing war production and imminent invasion. One of the main Allied 'black' stations was Gustav Siegfried Eins on which 'der Chef' (the Chief), apparently an old guard military veteran, launched into profanity-laden tirades against the upper echelons of the Nazi party. There was also 'grey' propaganda, essentially similar in content to 'black' but without a plausible explanation for its source - a 'black' radio station might broadcast on shortwave frequencies that dissidents could have access to, whereas a station on a powerful medium wave transmitter would be 'grey'.
British propaganda, from 1941, fell under the Political Warfare Executive (PWE). Prior to that the BBC was under the Ministry of Information while covert propaganda had been under various areas including the Secret Intelligence Service and Special Operations Executive. The head of 'black' propaganda, responsible for 'der Chef', was Sefton Delmer, born in Berlin to Australian parents and later a journalist in Germany. Delmer accompanied Hitler on his 1932 election campaign, where he learnt that Hitler's inner circle referred to him as 'der Chef', giving him the name for the announcer of Gustav Siegfried Eins. Prior to his 'black' work Delmer also made German language broadcasts on the BBC in 1940, giving 'cheerful little talks ... full of teasing and derision for the would-be invaders across the water, to show them that we in Britain were a long way from having our tails down". One broadcast was in the form of an English lesson:
"We English, as you know, are notoriously bad at languages," said I "and so it will be best if you learn a few useful English phrases before visiting us. For your first lesson we will take: the Channel crossing. Now just repeat after me: Das Boot sinkt... the boat is sinking. Das Wasser is kalt... the water is cold. Sehr kalt... very cold."
The lesson culminated in forms of the verb "to burn", finishing off with "Der SS-Sturmführer brennt auch ganz schön... The SS Captain is also burning quite nicely", building on rumours that had been spread of anti-invasion defences that would set the sea alight (not entirely without foundation) .
Some further reading:
Haw-Haw And Radio War: Fake News In The 1940s, University of Sheffield History Matters
The Fake British Radio Show That Helped Defeat the Nazis, Smithsonian Magazine
The Political Warfare Executive, Covert Propaganda, and British Culture, University of Durham
Listening to the World, BBC Monitoring Collection at the Imperial War Museum
The Major Developments in Political Warfare Throughout the War, 1938-1945, Y. M. Streatfield
The Black Art: British Clandestine Psychological Warfare Against the Third Reich, Lee Richards