Describe how heat is transferred from the core through the mantle to the crust?

Conduction probably is the main method in the crust, but do not underestimate fluid convection. Every time you find a hot spring, then that is an example of fluid convection ... Since these two are separate heat transfer from the outer core to the mantle is also mainly by conduction

Conduction and fluid convection transfers heat through mantle to the crust.