The Bombers have beaten the Crows in the race for Edwards, in a coup for the club after its disastrous 2013.
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Edwards, who is on tour with the Indigenous All-Stars in Ireland has played 12 games while Aylett, who is from NSW but arrived at the Giants via North Ballarat, has played just one game.
Aylett has had a history of knee injuries but is a talented half-back flanker.
It is a good result for Essendon who has managed to secure two talented youngsters for a third-round draft pick and could also receive a second round draft pick if it accepts the Western Bulldogs' offer of pick No.26 for key forward Stewart Crameri.
The Giants have managed to clear some salary cap space – with Aylett contracted with the club for 2014 – and two extra spots on its list as it attempts to balance the list with a mix of experienced players and talented youngsters.
The Giants are in the process of securing Jed Lamb from the Sydney Swans and remain in negotiations with Collingwood to secure Heath Shaw. It managed to snag Shane Mumford from the Sydney Swans for pick No.35.
It traded Dom Tyson and pick No.9 to Melbourne to secure pick No.2 and No.22 and it is likely to lose Taylor Adams to the Magpies.