Jordan asked the Saddam Hussein to extradite Zarqawi so they could try him for his role in the Millennium plot to bomb sites in Jordan including a hotel frequented by Americans [and Israelis too, IIRC].
Saddam refused the request.
After treatment [and after his cell in Baghdad decapitated a nun] he traveled north. In late summer 2002 he was in Syria where he had a training camp and it was the Syrian cell who assassinated Foley in October 2002.
All this before the US invasion of Iraq.
Jordan asked Syria to extradite him; like Iraq, Syria refused.
Zarqawi formed a pipeline in Syria to funnel recruits from Europe into Syria and later into Iraq where with Mullah Krekar’s group they eventually merged into al Dhouri’s group of Baath Party operatives. They became al Qaeda in Iraq after the US invasion.