It is not difficult to see that $Z(Inn(G)) \leq C_{Aut_{c}(G)}(Z(G)) \leq Aut_{c}(G)$ (see "FINITE GROUPS WITH CENTRAL AUTOMORPHISM GROUP OF MINIMAL ORDER"). Also we can check that if C_{Aut_{c}(G)}(Z(G)) wants to be a subgroup of $Inn(G)$ it should be equal to Z(Inn(G)) ... Thus I can ask my question in this way "If Z(Inn(G)) = C_{Aut_{c}(G)}(Z(G)) then what can I say about $G$?"