Step 1: Coagulation runs along two arms. The intrinsic arm starts inside the vessel from contact activation (Factor XII), and the extrinsic arm starts from outside the vessel after tissue injury.
Step 2: Injured tissue releases tissue factor, which is named Factor III or tissue thromboplastin. This is the spark for the extrinsic pathway.
Step 3: Tissue factor joins Factor VII and calcium ions to switch on Factor X. From Factor X onward both arms share the common pathway, so Factor X is convergence rather than the extrinsic starter, and Factors XI and XII sit in the intrinsic arm.
Step 4: Hence the molecule that activates the extrinsic system is Factor III.
\[\boxed{\text{Factor III (tissue factor)}}\]