  Let $(G_n)_{n \in \mathbb{N}}$ be a sequence of groups equipped with a
$d$-ary cloning system and denote by $\mathscr{T}_d(G_*)$ the resulting
Thompson-like group. In previous work joint with Zaremsky, we obtained
structural results concerning the group von Neumann algebra of
$\mathscr{T}_d(G_*)$, denoted by $L(\mathscr{T}_d(G_*))$. Under some natural
assumptions on the $d$-ary cloning system, we proved that
$L(\mathscr{T}_d(G_*))$ is a type $\text{II}_1$ factor. With a few additional
natural assumptions, we proved that $L(\mathscr{T}_d(G_*))$ is, moreover, a
McDuff factor. In this paper, we further analyze the structure of
$L(\mathscr{T}_d(G_*))$, in particular the inclusion $L(F_d) \subseteq
L(\mathscr{T}_d(G_*))$, where $F_d$ is the smallest of the Higman--Thompson
groups. First, we prove that if the $d$-ary cloning system is "diverse", then
the inclusion $L(F_d) \subseteq L(\mathscr{T}_d(G_*))$ is irreducible, a
considerable improvement of the result that $L(\mathscr{T}_d(G_*))$ is a type
$\text{II}_1$ factor. This allows us to compute the normalizer of $L(F_d)$ in
$L(\mathscr{T}_d(G_*))$, which turns out to be "trivial" in the diverse case,
implying that the inclusion is also singular. Then we prove that the inclusion
$L(F_d) \subseteq L(\mathscr{T}_d(G_*))$ satisfies the weak asymptotic
homomorphism property, which yields another proof that the inclusion is
singular. Finally, we finish the paper with an application: Using
irreducibility, our conditions for when $L(\mathscr{T}_d(G_*))$ is a type
$\text{II}_1$ McDuff factor, and the fact that $F_d$ is character rigid (in the
sense of Peterson), we prove that the groups $F_d$ are McDuff (in the sense of
Deprez-Vaes).
