TITLE: Groups which maintain all their subgroups’ automorphisms as inner automorphisms
QUESTION [15 upvotes]: Are there any groups, finite or infinite, other than the first three symmetric groups which maintain all their subgroups’ automorphisms as inner automorphisms (every automorphism of every subgroup extends to an inner automorphism of the whole group)?

REPLY [4 votes]: Yes. By this paper of Minasyan one can construct a finitely generated group $G$ with all proper subgroups cyclic of prime order $p\gg 1$, two conjugacy classes and trivial $Out(G)$. This group obviously satisfies the conditions of OP.