Summary: We show that any regular (right) Schreier extension of a monoid \(M\) by a monoid \(A\) induces an abstract kernel \(\Phi\colon M\to\frac{\operatorname{End}(A)}{\operatorname{Inn}(A)}\). If an abstract kernel factors through \(\frac{\operatorname{SEnd}(A)}{\operatorname{Inn}(A)}\), where \(\operatorname{SEnd}(A)\) 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 \(\Phi\). An abstract kernel \(\Phi\colon M\to\frac{\operatorname{SEnd}(A)}{\operatorname{Inn}(A)}\) (resp. \(\Phi\colon M\to\frac{\operatorname{Aut}(A)}{\operatorname{Inn}(A)})\) 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 \(\Phi\colon M\to\frac{\operatorname{SEnd}(A)}{\operatorname{Inn}(A)} \) (resp. \(\Phi\colon M\to\frac{\operatorname{Aut}(A)}{\operatorname{Inn}(A)})\), when it is not empty, is in bijection with the second cohomology group of \(M\) with coefficients in \(U(Z(A))\).