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Let N and H be groups, and here for a homomorphism f: H --> Aut(N) =...

Let N and H be groups, and here for a homomorphism f: H --> Aut(N) = group automorphism,

let N x_f H be the corresponding semi-direct product. Let g be in Aut(N), and  k  be in Aut(H),  Let C_g: Aut(N) --> Aut(N) be given by

conjugation by g. 
Now let  z :=  C_g * f * k: H --> Aut(N), where * means composition.

Show that there is an isomorphism
from Nx_f H to Nx_z H, which takes the natural copies of N, and H in each group to themselves.  

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