  A long-standing conjecture asserts that every finite nonabelian $p$-group has
a non-inner automorphism of order $p$. In this paper we prove the conjecture
for finite $p$-groups of coclass $4$ and coclass $5$ ($p\ge 5$). We also prove
the conjecture for an odd order nonabelian $p$-group $G$ with cyclic center
satisfying $Z_3(G)\cap C_G(G^p\gamma_3(G))\le Z(\Phi(G))$.

Group Theory