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Let $G$ be a finite 2-group of nilpotency class two such that $\frac{G}{Z(G)}=\{Z(G), aZ(G), bZ(G), abZ(G)\}\simeq C_{2}\times C_{2}$. Then do there exist a non inner automorphism $\alpha$ of $G$ such that $\alpha(a)\neq a$, $\alpha(b)\neq b$ and $\alpha(ab)\neq ab$ ? For example this is true for $D_{8}$, dihedral group of order 8, or $Q_{8}$, generalized quaternion group of order 8.

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    The meaning of "acts trivial only on $Z(G)$" is really not clear. Why not write what you mean in mathematical language? Do you perhaps mean "Does there exists a non-inner automorphisms $\alpha$ of $G$ such that $\{g \in G \mid \alpha(g)=g \} = Z(G)$?" ?2012-05-29
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    @DerekHolt: This is a follow-on from a comment I left to an [earlier question](http://math.stackexchange.com/questions/150632/on-automorphisms-group-of-some-finite-2-groups#comment346988_150632) of his (or, indeed, hers).2012-05-29
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    Instead of re-posting your question, you could [take it to the pro's](http://mathoverflow.net/questions/ask)?2012-06-06
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    For future reference: instead of posting a new version of your question, you should instead edit the old version to improve it based on the comments posted. I've merged the old version into this one, since it appears you are providing more information to address @DerekHolt's comment.2012-06-06
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    This is almost different with earlier question.2012-06-07
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    @user1729: I'd call Derek a pro.2012-06-07

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