Finite groups with its power automorphism groups having small indices

In this paper, a finite group G with |Aut(G): P(G)| = p or pq is determined, where P(G) is the power automorphism group of G, and p, q are distinct primes. Especially, we prove that a finite group G satisfies |Aut(G): P(G)| = pq if and only if Aut(G)/P(G) }~ S 3. Also, some other classes of finite groups are investigated and classified, which are necessary for the proof of our main results.