We aim to provide the first data-constrained estimate of the metal-free (Population III; Pop III) star formation rate density $dot{rho}_{*}^{III}$ required at high-redshifts ($z gtrsim 16$) in order to reproduce both the amplitude and the redshift of the EDGES 21-cm global signal. Our model accounts for the Lyman Alpha (Ly$alpha$), radio and X-ray backgrounds from both Pop III and metal-enriched Population II (Pop II) stars. For the latter, we use the star formation rate density estimates (and the Ly$alpha$ background) from the {it Delphi} semi-analytic model that has been shown to reproduce all key observables for galaxies at $z gtrsim 5$; the radio and X-ray backgrounds are fixed using low-$z$ values. The constraints on the free parameters characterizing the properties of the Pop III stars are obtained using a Markov Chain Monte Carlo analysis. Our results yield a $dot{rho}_{*}^{III}$ that whilst increasing from $z sim 21-16$ thereafter shows a sharp decline which is in excellent agreement with the results found by citet{valiante2016} to simulate the growth of $z sim 6 - 7$ quasars and their host galaxies, suggesting that the bulk of Pop III star formation occurs in the rarest and most massive metal-poor halos at $z lesssim 20$. This allows Pop III stars to produce a rapidly growing Ly$alpha$ background between $z sim 21-15$. Further, Pop III stars are required to provide a radio background that is about $3-4$ orders of magnitude higher than that provided by Pop II stars although Pop II stars dominate the X-ray background.
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