Lifting functors from to

Let F (correspondingly P) denote the abelian category of functors (strict polynomial functors in the sense of Friedlander and Suslin) from finite dimensional vector spaces over Fp to vector spaces over Fp. These two categories are related via the exact forgetful functorι:P→F.The category F is strongly related to topology and representation theory of symmetric and general linear groups but the homological algebra in F is rather mysterious. The category P is easier for cohomological calculations. The known ExtF(.,.) calculations are obtained only for functors which belong to the image of ι and are performed using comparison of ExtP- and ExtF-groups induced by ι. The aim of the following note is to find cohomological conditions which guarantee that a given functor F∈F comes from P via ι.