The following problem is considered: what kind of finite groups can function as fullautomorphism group of a finite group? We first show that if the finite group K has a normalSylow p-subgroup such that |K/Z(K)|p=p2, then K has an automorphism of order 2. Usingthis result, we have shown that if G is an odd order group with order psm (1 ≤s ≤3), wherep is the smallest prime divisor of |G|, p m and m is cubefree, then G cannot function as fullautomorphism group.
Some Groups of Odd Order Which Cannot Function as Automorphism Groups. Acta Mathematica Sinica, Chinese Series, 1996, 39(4) https://doi.org/10.12386/A1996sxxb0077