SCHWEITZER 1979 Problem 2

Let $\Gamma$ be a variety of monoids such that not all monoids of $\Gamma$ are groups. Prove that if $A \in \Gamma$ and $B$ is a submonoid of $A$ , there exist monoids $S \in \Gamma$ and $C$ and epimorphisms $\varphi : S \rightarrow A, \;\varphi_1 : S \rightarrow C$ such that $((e)\varphi_1^{-1})\varphi=B$ ( $e$ is the identity element of $C$ ).

L. Marki