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My question is motivated by the following two posts

On finite 2-groups that whose center is not cyclic

and

Automorphisms of group extensions

Question: Assume that $A,B,C$ are there algebraic structures (groups, algebra, banach or $C^{*}$ algebra). Let we have an extension: $$0\to A \to B \to C \to 0$$

Assume that $A$ is commutative and every automorphism of $A$ has an extension to an automorphism of $B$. Does it imply that $A\subset Z(B)$, the center of $B$?

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No. For example, $S_3$ contains a characteristic, abelian subgroup not contained in the centre of the group. Both of the automorphisms of this subgroup extend to (inner) automorphisms $G$.

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