Permutational wreath pullbacks and framed braid-type groups

Let $\sigma\colon G \to S_n$ be a surjective homomorphism and let $H$ be a group. We introduce the \emph{permutational wreath pullback} $$H \wr_\sigma G = H^n \rtimes_\sigma G,$$ where the action of $G$ on $H^n$ is induced by permutation of coordinates via $\sigma$, and undertake a systematic structural study of this construction. We determine the center and the abelianization in full generality. We further show that $H \wr_\sigma G$ admits a natural interpretation as the pullback of the classical wreath product $H \wr S_n$ along $\sigma$, providing a conceptual explanation for its functorial behavior. When $H$ is finitely generated abelian, we establish a criterion for the abelian kernel $H^n$ to be characteristic and for $H \wr_\sigma G$ to inherit the $R_\infty$-property from $G$; we verify this criterion for kernels arising from the virtual braid group $VB_n$ and the virtual twin group $VT_n$, obtaining new families of framed groups with the $R_\infty$-property. Rigidity results show that the abelian kernel, $n$, $H$, and $G$ are determined by the abstract group $H \wr_\sigma G$. Applications include uniform descriptions of classical, surface, virtual, and singular framed braid groups, and a reduction of splitting problems for framed surface braid groups to the classical Fadell--Neuwirth setting.