A note on the $R_\infty$ property for groups $\mathrm{FAlt}(X)\leqslant G\leqslant \mathrm{Sym}(X)$

Given a set $X$, the group $\mathrm{Sym}(X)$ consists of all bijections from $X$ to $X$, and $\mathrm{FSym}(X)$ is the subgroup of maps with finite support i.e. those that move only finitely many points in $X$. We describe the automorphism structure of groups $\mathrm{FSym}(X)\le G\le \mathrm{Sym}(X)$ and use this to state some conditions on $G$ for it to have the $R_\infty$ property. Our main results are that if $G$ is infinite, torsion, and $\mathrm{FSym}(X)\le G\le \mathrm{Sym}(X)$, then it has the $R_\infty$ property. Also, if $G$ is infinite and residually finite, then there is a set $X$ such that $G$ acts faithfully on $X$ and, using this action, $\langle G, \mathrm{FSym}(X)\rangle$ has the $R_\infty$ property. Finally we have a result for the Houghton groups, which are a family of groups we denote $H_n$, where $n \in \mathbb{N}$. We show that, given any $n\in \mathbb{N}$, any group commensurable to $H_n$ has the $R_\infty$ property.