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On automorphism of some finite 2-group of class nilpotency two

Let $G$ be a finite 2-group of nilpotency class two such that $\frac{G}{Z(G)}\simeq C_{2}\times C_{2}$. Then do there exist a non inner automorphism of $G$ that acts trivial only on $Z(G)$? For example $D_{8}$, dihedral group of order 8, has a non inner automorphism that acts trivial only on $Z(G)$.

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