Let $G$ be a finite group. Recall that a family of subgroups of $G$, denoted by $\mathcal{F}$, is a collection of subgroups of $G$ that is closed under taking subgroups and conjugation.

Denote by $\Sigma_m$ the group of permutations of the set m$=\{1,2,\ldots,m\}$, that is the symmetric group of degree $m$. Let $\mathcal{F}^{G,\Sigma_m}$ be the family of subgroups $L\leq G\times \Sigma_m$ such that $L\cap \{e\}\times \Sigma_m= \{(e,e)\}.$ The projection function $\mathrm{pr}_1:G\times\Sigma_m\to G$ has kernel \begin{align} \mathrm{ker}(\mathrm{pr}_1)&=\{(g,\sigma)\in G\times\Sigma_m | \ \mathrm{pr}_1(g,\sigma)=e\} \\ &=\{(e,\sigma)|\ \sigma\in\Sigma_m\} \\ &=\{e\}\times\Sigma_m \end{align}

So, a subgroup $L\leq G\times\Sigma_m$ belongs in the family $\mathcal{F}^{G,\Sigma_m}$ if and only if the restriction to $L$ of the projection, i.e., $\mathrm{pr}_1|_{L}: L\to G$, is an injection.

There is a bijective corresponce between the family $\mathcal{F}^{G,\Sigma_m}$ and the set of all representations, of all subgroups of $G$, on the finite set m$=\{1,\ldots, m\}$, that is, all group homomorphisms $\phi:H\to \Sigma_m$ for every $H\leq G.$

My question is : Is there a subfamily of the family $\mathcal{F}^{G,\Sigma_m}$ that contains precisely the actions of the group $G$ on m, that is, group homomorphisms $\phi:G\to \Sigma_m$ and for every subgroup $H\leq G$ with inlcusion $i:H\to G$ we only allow restrictions of group actions of $G$ to $H$, that is, we only allow $i^*\phi: H\to \Sigma_m$ ?

Any input would be helpful


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