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Let $m\leqslant n\in \mathbb {N}$
, and $G\leqslant \operatorname {Sym}(m)$
and $H\leqslant \operatorname {Sym}(n)$
. In this article, we find conditions enabling embeddings between the symmetric R. Thompson groups ${V_m(G)}$
and ${V_n(H)}$
. When $n\equiv 1 \mod (m-1)$
, and under some other technical conditions, we find an embedding of ${V_n(H)}$
into ${V_m(G)}$
via topological conjugation. With the same modular condition, we also generalize a purely algebraic construction of Birget from 2019 to find a group $H\leqslant \operatorname {Sym}(n)$
and an embedding of ${V_m(G)}$
into ${V_n(H)}$
.
