surjection or epimorphism?

Suppose G is a group and G_(i+1), G_i are two subgroups such that G_(i+1) is
normal in G_i. Let f:G->H be a surjection between the groups G and H.
If f(x_(i+1)) in G(x_(i+1)) and f(x_i) in G_i then
f(x_i).f(x_(i+1)).f(x_i)^(-1) = f(x_i.x_(i+1).x_i^(-1)) is in f(G_i),
because G_(i+1) is normal in G_i.

Is this statement true as it is written (that is, even if f is not a
homomorphism)?
TIA


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