A note on noninner automorphisms of order $p$ for finite $p$-groups of coclass 2

In this note, the existence of noninner automorphisms of order 2 for finite 2-groups of coclass 2 is proved. Combining our result with a recent one due to Y. Guerboussa and M. Reguiat (see arXiv:1301.0085), we prove that every finite $p$-group of coclass 2 has a noninner automorphism of order $p$ leaving the center elementwise fixed.