Unipotent automorphisms of modular group algebras and the homology of the Lazard Lie algebra

Let $p$ be an odd prime and $H$ a finite $p$-group. Hertweck and Soriano parametrize the central Frattini extensions of $H$ by subspaces of a relation module $V$, matching isomorphism classes of extensions with orbits of $\mathrm{O}\mathrm{u}\mathrm{t}(H)$ and isomorphism classes of the associated small group rings with orbits of $\mathrm{O}\mathrm{u}\mathrm{t}({\mathbb{F}}_{p}H)$. The discrepancy between the two is measured by the group $U(H)\le GL(V)$ of displacements induced by the normalized unipotent automorphisms of ${\mathbb{F}}_{p}H$; it is the mechanism behind the counterexamples to the modular isomorphism problem in characteristic $2$. We prove that for $H$ of exponent $p$ and class at most $p-2$ every such automorphism acts on $V$ exactly as an automorphism of $H$ does, and that $U(H)=\exp \mathrm{\Omega }({\mathrm{D}\mathrm{e}\mathrm{r}}_{>0}(\mathfrak{l}(H)))$, where $\mathfrak{l}(H)$ is the Lazard Lie algebra of $H$, ${\mathrm{D}\mathrm{e}\mathrm{r}}_{>0}$ its derivations raising the lower central filtration, and $\mathrm{\Omega }$ the natural action of derivations on $H_2(\mathfrak{l}(H))$. Hence $U(H)=1$ precisely when every derivation of positive degree acts trivially on $H_2(\mathfrak{l}(H))$. We compute $U(H)$ for all groups of class $2$ and exponent $p$, obtaining $p^{2n(n-1)}$ for the Heisenberg groups $\mathrm{H}\mathrm{e}\mathrm{i}\mathrm{s}(p^n)$.