When is Inn(G) abelian or cyclic.

Question

I proved Inn(G) cyclic implies G abelian.

and if G finite group, then Aut(G) cyclic iff G cyclic and order G is either 1,2, 4 or $p^k$ or 2$p^k$.

Is there chacterisation for(given G is finite group):

Question 1: When Inn(G) is cyclic?

Question 2: When Inn(G) is abelian?

Question 3: When Aut(G) is abelian?