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?
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: 

