When is Inn(G) abelian or cyclic.

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 pkp^k or 2pkp^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?

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Question 1: see here. Question 2: see here. Question 3: see here.
– Dietrich Burde
2 days ago

  

 

@DietrichBurde thanks but for question 2 atleast you only mentioned sufficinet condition for Inn(G) abelian. Is there a necessary condition also
– Sushil
2 days ago

  

 

Inn(G)\textrm{Inn}(G) is abelian iff G/Z(G)G/Z(G) is abelian iff GG is at most 22-step nilpotent.
– Keith Kearnes
yesterday

  

 

sorry not familiar with term 2-step nilpotent. Can you explain or give soem reference @KeithKearnes
– Sushil
yesterday

  

 

en.wikipedia.org/wiki/Nilpotent_group
– Keith Kearnes
18 hours ago

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