Induction and restriction of characters and Hall subgroups

Let H be a Hall π-subgroup of a π-separable group G. Given a character χ∈Irr(G), we determine when it happens that χ=αG for every irreducible constituent α of χH, and dually, given a character α∈Irr(H), we determine when it happens that α=χH for every irreducible constituent χ of αG. We also study some related questions.