The following question is for my own curiosity as I take some time to get reacquainted with group theory.

Let G be a semi-direct product of the groups N and K with multiplication defined by the automorphism $\phi$ from K to Aut(N). Let Fix($\phi$) be the set of all elements of N that are mapped to themselves by all elements of the range of $\phi$. Clearly every element of Fix($\phi$) commutes with all elements of K and every element of the kernel of $\phi$ commutes with every element of N.

If Fix($\phi$) is the trivial group in N and Ker($\phi$) is the trivial group in K, does that imply that the center of G is trivial? If so, could someone point me to a reference or proof. If not, then a counter example.