Answer to Question #18547 in Abstract Algebra for sanches

Question

For any field k and for any normal subgroup H of a group G, assume further that [G : H] is finite and prime to char k. Let V be a kG-module and W be a kH-module. Show that W is a semisimple kH-module iff the induced module kG ⊗kH W is a semisimple kG-module.