The formation processes that led to the current Galactic stellar halo are still under debate .
Previous studies have provided evidence for different stellar populations in terms of elemental abundances and kinematics , pointing to different chemical and star-formation histories .
In the present work we explore , over a broader range in metallicity ( -2.2 < [ Fe / H ] < +0.5 ) , the two stellar populations detected in the first paper of this series from metal-poor stars in DR13 of the Apache Point Observatory Galactic Evolution Experiment ( APOGEE ) .
We aim to infer signatures of the initial mass function ( IMF ) and the star-formation history ( SFH ) from the two \alpha -to-iron versus iron abundance chemical trends for the most APOGEE-reliable \alpha -elements ( O , Mg , Si and Ca ) .
Using simple chemical-evolution models , we infer the upper mass limit ( M _ { up } ) for the IMF and the star-formation rate ( SFR ) , and its duration for each population .
Compared with the low- \alpha population , we obtain a more intense and longer-lived SFH , and a top-heavier IMF for the high- \alpha population .