Symmetric forcing systems, supports, and hereditarily symmetric names

A symmetric system $(P,G,\mathcal{F})$ consists of a forcing preorder, a group $G$ of its automorphisms, and a normal filter $\mathcal{F}$ of subgroups of $G$. Put $\mathrm{sym}(\dot{x})=\{\pi \in G:\pi \dot{x}=\dot{x}\}$. A name is symmetric if its stabilizer lies in $\mathcal{F}$, and hereditarily symmetric if it is symmetric and every subname occurring in it is hereditarily symmetric. Write ${\mathrm{H}\mathrm{S}}_{\mathcal{F}}$ for these names. A subgroup $H$ supports $\dot{x}$ when $H\in \mathcal{F}$ and $H\subseteq \mathrm{sym}(\dot{x})$. In a coordinate presentation with pointwise stabilizers, a finite coordinate set $E$ supports $\dot{x}$ when ${\mathrm{fix}}_G(E)\in \mathcal{F}$ and ${\mathrm{fix}}_G(E)\subseteq \mathrm{sym}(\dot{x})$.

Normality and $\mathrm{sym}(\pi \dot{x})=\pi \mathrm{sym}(\dot{x}){\pi }^{-1}$ give $\pi \mathrm{‘}\mathrm{‘}\mathrm{H}\mathrm{S}=\mathrm{H}\mathrm{S}$. If $G_0$ is a ground-model generic filter—distinct from the automorphism group $G$—the symmetric interpretation is ${\mathrm{H}\mathrm{S}}_{\mathcal{F}}^{G_0}=\{{\dot{x}}_{G_0}:\dot{x}\in {\mathrm{H}\mathrm{S}}_{\mathcal{F}}\}$. The hereditary clause makes this class transitive after evaluation. No AC is assumed.