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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?

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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).

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  • $\begingroup$ Thank you for the link. So, seemingly the following chain holds: $L(G)\le Z(G)\le H(G)\le G$, thus hinting for possibly further quotients than $\operatorname{Inn}(G)\cong G/Z(G)$ and $\operatorname{Inn}_c(G)\cong H(G)/Z(G)$? (+1) and accepted. $\endgroup$
    – user1007416
    Mar 8, 2022 at 12:52
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    $\begingroup$ Yes that is correct. And both centralizers $\rm{Aut}_c(G)=C_{\rm{Aut}(G)}(\rm{Inn}(G))$ and $\rm{Inn}_c(G)=C_{\rm{Inn}(G)}(\rm{Aut}(G))$ are interesting objects to study. $\endgroup$ Mar 8, 2022 at 14:25

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