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This question already has an answer here:

Any hints for proving that innG $\triangleleft$ autG for any group G.

where innG(Inner automorphism of G) and autG(Automorphism of G) and H $\triangleleft$ K (H is normal subgroup of K).

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marked as duplicate by Dietrich Burde abstract-algebra May 2 '17 at 18:58

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Suppose you have $\tau \in \text{inn}(G)$. Then there is a $g \in G$ such that $\tau(h) = ghg^{-1}$ for all $h \in G$. If $\sigma \in \text{Aut}(G)$, can you express $\sigma\tau\sigma^{-1}(h)$ as an inner automorphism (by an element related to $g$ somehow)?

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