Locally finite Suzuki–Higman $2$-groups

We prove the following:
THEOREM. Let $U$ be a locally finite Suzuki–Higman $2$-group with respect to an automorphism group $H$. Then $U$ and $H$ are representable as the respective unions of ascending chains of finite subgroups
$$\begin{align*} U_1<U_2<&\dots<U_n<\dots,\\ H_1<H_2<&\dots<H_n<\dots, \end{align*}$$
in which case every subgroup $U_n$ is a Suzuki $2$-group with respect to $H_n$.