If $G$ is a (finite) group in which every subgroup is characteristic, then it's easy to show that $G$ is cyclic (and conversely of course).
A more interesting generalization is to instead consider groups in which every conjugacy class of subgroups is characteristic. In other words, for any automorphism $\phi\colon G\to G$ and any subgroup $H\leqslant G$ there exists some $g\in G$ such that $\phi(H) = H^g$.
This class of groups is much larger. For instance, any group with trivial outer automorphism group satisfies this property. But there are also examples like $SL_2(\mathbb{F}_3)$ and $SL_2(\mathbb{F}_5)$ which have non-trivial outer automorphisms but every conjugacy class of subgroups is characteristic.
I doubt there's anything like a classification of groups of this form since it's already not feasible to classify groups with trivial automorphism group. But these groups have cropped up in my research and I'd like to at least know:
(1) Is there an established name for groups of this form?
(2) Are there any papers that consider them?
There are also some similar stronger properties that I would be equally interested in, like if any two isomorphic subgroups are conjugate or any two subgroups of equal order are conjugate (the above examples of $SL_2(\mathbb{F}_3)$ and $SL_2(\mathbb{F}_5)$ both satisfy these stronger properties).
Best Answer
Let $G$ be a group, $\alpha \in Aut(G)$ is called intense if $\alpha$ sends each subgroup of $G$ to a conjugate. The set of all intense automorphisms $Int(G)$ is a subgroup of $Aut(G)$ and is in fact a normal subgroup. There is a natural connection between intense automorphisms and Galois cohomology. The intense automorphisms were studied by Mima Stanojkovski, who wrote her PhD thesis (Leiden, 2017) entitled Intense Automorphisms of Finite Groups under the direction of Hendrik Lenstra. See also here. Many results on the structure of $Int(G)$ can be found in her thesis, especially for (finite) $p$-groups $G$ (including a result on infinite pro-$p$-groups).