On the coincidence of extremal and strong epimorphisms when $(Epi,Mono)$ is a factorization system

Question

Suppose that $(Epi,Mono)$ is a factorization system on a category $\mathfrak{A}$. It may be easily shown that every epimorphism is extremal. On the other hand, in every category, the implication strong epimorphism $\implies$ extremal always holds. Is it possible to say something on the reverse implications when $(Epi,Mono)$ is a factorization system on $\mathfrak{A}$. If not, which can be a possible counterexample?