On faithful irreducible projective representations of finite groups over a field of characteristic p

Let K be a field of finite characteristic p and G a finite group with a normal Sylow p-subgroup. We give necessary and sufficient conditions for G to have a faithful irreducible projective representation over K. In the case when G is an abelian p-group and K is not a perfect field we find also all dimensions of faithful irreducible projective representations of G over K and show that the group G has infinitely many isomorphism classes of faithful irreducible projective representations of each possible dimension d≠1.