P-extensible implies inner

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Suppose p is a prime number and P is a p-group, i.e., a group in which the order of every element is a power of p. Suppose \sigma is an automorphism of P such that for any p-group Q containing P, there is an automorphism \sigma' of Q whose restriction to P equals \sigma. Then, \sigma is an Inner automorphism (?) of P.


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