If H has trivial center. I know that if Z(G) is trivial, then Aut(G) has trivial center, but I need help understanding what is happening when G has a trivial center and Aut(G) ≈ Inn(G).

I know that when Z(G) is trivial, then G ≈ Inn(G) but when does Aut(G) ≈ Inn(G) imply?