Closed subgroups and submonoids of pointwise convergence.

Question

Considering the topological group $\operatorname{Sym}(\mathbb{N})$ with the topology of pointwise convergence given by:

1. $g$ is in the closure of $F$ if for all finite $A \subsetneq \mathbb{N}$ there exists an $f \in F$ that agrees with $g$ on $A$.
2. Say that $F$ is closed if it is equal to its own closure.

Let $\langle F \rangle$ be the smallest closed subgroup of $\operatorname{Sym}(\mathbb{N})$ containing $F$. Let $\operatorname{cl}_{tm}(F)$ be the smallest closed monoid of $\mathbb{N}^\mathbb{N}$ containing $F$.

I have seen in texts the following claim:

$\langle F \rangle \subseteq \operatorname{cl}_{tm}(F)$, specifically $\langle F \rangle = \operatorname{cl}_{tm}(F) \cap \operatorname{Sym}(\mathbb{N})$.

I do not understand how this is the case and would appreciate some help explaining.