$\operatorname{Aut}_c(G)$ vs "$\operatorname{Inn}_c(G)$"

Question

As an application of the correspondence theorem, time ago I saw that the group of the inner automorphisms which commute with every automorphism -say $\operatorname{Inn}_c(G)$- is isomorphic to $H(G)/Z(G)$, where: $$H(G)=\{g \in G \mid \varphi(g)g^{-1} \in Z(G), \forall \varphi \in \operatorname{Aut}(G) \}\supseteq Z(G)$$ More recently, I've come across with the group of the automorphisms which commute with every inner automorphism. This central automorphism group, $\operatorname{Aut}_c(G)$, has much more literature than $\operatorname{Inn}_c(G)$, which hasn't got any indeed (at least to my knowledge). Is $\operatorname{Inn}_c(G)$ of any interest and hence mentioned somewhere?

Answer 1

Good question. The group of automorphisms you are looking at is basically $\rm{Inn(G)} \cap Z(\rm{Aut}(G))$, where the center is taken in $\rm{Aut(G)}$. These are called autocentral automorphisms. These have been studied in more detail amongst others by (the students of) prof. M.R. Moghaddam (see for example here).