Preimage of a subgroup

Suppose Ο•:Gβ†’Gβ€² is a surjective homomorphism, H≀G is a subgroup containing ker⁑(Ο•), and Hβ€²=Ο•(H). Prove Ο•βˆ’1(Hβ€²)=H, where Ο•βˆ’1(Hβ€²)={g∈Gβˆ£Ο•(g)∈Hβ€²}. Make sure to state explicitly where each hypothesis is used.