We show that, with any admissible abstract kernel of monoids, which is a monoid homomorphism \( \Phi\colon M \to\frac{End(A)}{Inn(A)} \), where both \( M \) and \( A \) are monoids and \( \Phi \) satisfies a suitable condition, it is possible to associate 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 \( A \), whose \( M \)-module structure is determined by \( \Phi \). We show that the obstruction is zero precisely when the admissible abstract kernel is induced by a regular Schreier extension of monoids. We equip the set of equivalence classes of admissible abstract kernels inducing the same action of \( M \) on \( U(Z(A)) \) with a commutative monoid structure. We prove that the quotient of this monoid with respect to the submonoid of equivalence classes of admissible abstract kernels induced by a regular Schreier extension is isomorphic to the third cohomology group \( H^3(M,U(Z(A))) \).