  Building upon previous results, a classification is given of finite
$p$-groups of which subgroups of order $p$ are all fused. This completes the
classification problem dated back to Higman 1963 on the so-called Suzuki
$2$-groups, and confirms a conjecture of Gross proposed in 1974. As a
consequence, two open problems on AT-groups and FIF-groups are solved.
