A note on non-inner automorphism conjecture

In this paper we prove that every $2$-generator finite $p$-group $G$ has a non-inner automorphism of order $p$ leaving $G^p{\gamma }_4(G)$ elementwise fixed ($p\ge 5$). Moreover, we prove a $2$-generator finite $3$-group satisfying $\mathrm{\mid }{\mathrm{\Omega }}_1(Z_2(G))\mathrm{\mid }=p^2$ has a non-inner automorphism of order $p$ leaving $G^p{\gamma }_3(G)$ elementwise fixed. As a consequence we prove the non-inner automorphism conjecture for every finite $p$-group of coclass $4$ ($p\ge 3$), and coclass $5$ ($p\ge 5$).