We define the classes of strongly ω1-weak pω·2+n-projective, solidly ω1-weak pω·2+n-projective and nicely ω1-weak pω·2+n-projective abelian p-groups and study their crucial properties. This continues our recent investigations of this branch, published in Hacettepe J. Math. Stat. (2013) and Bull. Malaysian Math. Sci. Soc. (2014), respectively.