Let $\mathsf C$ be the category of topological monoids, that is, the category of monoids in $(\textsf{Top}, \times)$.

Can the model category structure on $\textsf{Top}$ (Serre fibrations, cofibrations, weak homotopy equivalence) be transferred to $\mathsf C$ along the free and forgetful pair of functors ?

What are the functorial factorizations in $\mathsf C$? Is there a cylinder object in $\mathsf C$?

I'm mostly interested in computing a homotopy pushout in the category $\mathsf C$, so any ideas how to do that would be helpful too.