We show that any regular (right) Schreier extension of a monoid M by a monoid A induces an abstract kernel .
If an abstract kernel factors through , where is the monoid of surjective endomorphisms of A, then we associate to it an obstruction, which is an element of the third cohomology group of M with coefficients in the abelian group of invertible elements of the center of A, on which M acts via Φ.
An abstract kernel (resp. ) is induced by a regular weakly homogeneous (resp. homogeneous) Schreier extension of M by A if and only if its obstruction is zero.
We also show that the set of isomorphism classes of regular weakly homogeneous (resp. homogeneous) Schreier extensions inducing a given abstract kernel (resp. ), when it is not empty, is in bijection with the second cohomology group of M with coefficients in .