Question:Inner automorphism pull back along quotient maps

Q: '''It is clear that an inner automorphism can be pulled back via any quotient map, i.e., an inner automorphism of a quotient arises from an inner automorphism of the original group. Are there outer automorphisms for which this is always true?'''

A: No. This is similar to the previous question, and is proved using similar methods. The direction you mention above is inner implies quotient-pullbackable. The other, harder, direction is quotient-pullbackable equals inner. For more discussion, see extensible automorphisms problem.