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I am looking for all finite groups $G$ such that for each subgroup $H$ of $G$ and each automorphism $\sigma$ of $H$ there exists an automorphism $\psi$ of $G$ whose restriction to $H$ is $\sigma$. Is there any reference on this problem?

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These groups are called the groups of injective type. It is known that nonabelian finite groups of injective type have even order.

See the following papers:
http://dx.doi.org/10.1017/S0017089512000031
http://dx.doi.org/10.1142/S0219498807002235

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  • $\begingroup$ You are welcome @majid-arezoomand $\endgroup$
    – Name
    Commented May 25, 2013 at 13:54

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