  In this paper we prove that for $p\ge 5$, every $2$-generator finite
$p$-group $G$ has a non-inner automorphism of order $p$ leaving
$G^p\gamma_4(G)$ elementwise fixed. As a consequence we have the same result
for finite $p$-groups of coclass $4$ and coclass $5$ for $p\ge 5$.
