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: 

