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We show that any regular (right) Schreier extension of a monoid M by a monoid A induces an abstract kernel Φ: M → End(A)/Inn(A) . If an abstract kernel factors through SEnd(A)/Inn(A) , where SEnd(A) 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 U(Z(A)) of invertible elements of the center Z(A) of A, on which M acts via Φ. An abstract kernel Φ: M → SEnd(A)/Inn(A) (resp. Φ: M → Aut(A)/Inn(A) ) 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 Φ: M → SEnd(A)/Inn(A) (resp. Φ: M → Aut(A)/Inn(A) ), when it is not empty, is in bijection with the second cohomology group of M with coefficients in U(Z(A)).