Abelian p-group reduction conjecture for automorphism groups

Let $\lambda>\aleph_0$, let $G\in\mathbf{K}^{\lambda}_{\mathrm{lf}}$ be an abelian $p$-group, and let $H\leqslant G$ satisfy $|H|<\lambda$. Write $Aut_H(G)$ for the subgroup of automorphisms of $G$ that fix $H$ pointwise. A subgroup has power $\lambda$ when its cardinality is $\lambda$.

Abelian p-group reduction conjecture. $Aut_H(G)$ has a locally finite subgroup of power $\lambda$.

This is presented as a sufficient reduction for the preceding automorphism-group conjecture. The paper does not settle it for uncountable $\lambda$; the countable case of the broader assertion is proved separately.