Finite p-group with center of prime order and inner automorphism group maximal in p-Sylow-closure of automorphism group implies every p-automorphism is p-normal-extensible

Statement
Suppose $$P$$ is a finite $$p$$-group such that the center $$Z(P)$$ is a cyclic group of order $$p$$. Suppose, further, that the inner automorphism group $$\operatorname{Inn}(P)$$ is a maximal subgroup in the normal closure of any $$p$$-Sylow subgroup in the automorphism group $$\operatorname{Aut}(P)$$. Then, if $$\sigma$$ is a $$p$$-automorphism of $$P$$ (i.e., an automorphism whose order is a power of $$p$$), and $$Q$$ is a group containing $$P$$ as a normal subgroup, $$\sigma$$ can be extended to an automorphism $$\sigma'$$ of $$Q$$.

Examples
The prototypical example of this is where $$P$$ is dihedral group:D8. In this case, $$\operatorname{Aut}(P)$$ is also a dihedral group of order eight, with the inner automorphism group a Klein four-subgroup. Clearly, all the conditions for the statement are satisfied, and hence, all the $$2$$-automorphisms of the dihedral group (which in this case means all automorphisms) can be extended to any group containing this as a normal subgroup.

Related facts

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