because Z(D8) = {1,r^2}. so the center has to map to itself. an automorphism is bijective, meaning s can't map to r^2, because r^2 already is mapped to r^2. so we have exactly 4 choices:

s-->s

s-->rs

s-->r^2s

s-->r^3s

homomorphisms preserve commutativity, if ab = ba in G, then φ(a)φ(b) = φ(ab) = φ(ba) = φ(b)φ(a) in H. so φ(Z(G)) is always contained in Z(φ(G)).

if φ is an isomorphism, then φ(Z(G)) has to BE Z(φ(G)), if you have a bijective map f:A-->B then f(A) = B, right?