Every automorphism is center-fixing and outer automorphism group is rank one p-group implies not every normal-extensible automorphism is inner

Definition
Suppose $$G$$ is a group satisfying both the following conditions:


 * 1) $$\operatorname{Out}(G)$$, the outer automorphism group, is a nontrivial group having a unique minimal subgroup. In other words, it is a p-group of rank one. (when finite, it is either a cyclic group of prime power order or a generalized quaternion group).
 * 2) Every automorphism of $$G$$ is a center-fixing automorphism.

Then, the inverse image in $$\operatorname{Aut}(G)$$ of the unique minimal subgroup of $$\operatorname{Out}(G)$$ is contained in the group of normal-extensible automorphisms. In particular, not every normal-extensible automorphism is inner.

Related facts

 * Every automorphism is center-fixing and inner automorphism group is maximal in automorphism group implies every automorphism is normal-extensible
 * Centerless and maximal in automorphism group implies every automorphism is normal-extensible

Examples
Every finite cyclic group occurs as the outer automorphism group of a finite simple non-abelian group. More specifically, the cyclic group of order $$r$$ is the outer automorphism group of the projective special linear group of degree two $$PSL(2,2^r)$$. In particular, when $$r = p^k$$ for some prime number $$p$$ and $$k \ge 1$$, we obtain situations where the theorem applies.