Let $G$ be a finite 2-group of nilpotency class 2 such that $\frac{G}{Z(G)}\simeq C_{2}\times C_{2}$. I want information about its automorphisms group. Please guide me. Thank you
On automorphisms group of some finite 2-groups
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group-theory
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0Is this a homework question, or something else? A first example: [D_8](http://groupprops.subwiki.org/wiki/Dihedral_group:D8#Automorphisms_and_endomorphisms). If $G$ is a $2$-group such that $G/Z(G)\cong C_2\times C_2$ the automorphisms are actually quite interesting, as the inner automorphisms act trivially on $Z(G)$ and $G/Z(G)$ even though $Z(G)$ is characteristic (and $\operatorname{Inn}(G)\cong G/Z(G)$)! (I think, but cannot prove, that every outer automorphism is not hidden in this way, which would be quite a nice approach to your problem.) – 2012-05-28