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
Suppose is a finite -group such that the center is a cyclic group of order . Suppose, further, that the inner automorphism group is a maximal subgroup in the normal closure of any -Sylow subgroup in the automorphism group . Then, if is a -automorphism of (i.e., an automorphism whose order is a power of ), and is a group containing as a normal subgroup, can be extended to an automorphism of .
The prototypical example of this is where is dihedral group:D8. In this case, 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 -automorphisms of the dihedral group (which in this case means all automorphisms) can be extended to any group containing this as a normal subgroup.